Bertie3 Exercises. 1 Problem Set #3, PD Exercises

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Bertie3 Exercises 1 Problem Set #3, PD Exercises 1. 1 ( x)qx P 2 ( x)w x P ( x)(qx W x) 2. 1 ( x)(ax (Bx Cx)) P 2 Bd P Ad 3. 1 ( x)jxx P 2 ( y) Ry P ( y)(jyy Ry) 4. 1 ( x)(lx (Mx Bx)) P 2 ( y)(my Jy) P ( x)(lx Jx) 5. 1 ( w)( z)(bzw Kzw) P 2 Bcd P Kcd 1

6. 1 ( x)(ax Bx) P ( x)(bx Cx) (Ak Ck) 7. 1 ( x)jx ( y)ky P ( x)(jx ( y)ky) 8. 1 ( x)bx Be P ( x)bx Be 9. 1 ( x)(p x W x) P 2 P a P W a 10. 1 ( x)(f x Gx) P ( y)f y 11. 1 ( z)((f z Gzf) Hz) P ( x)gxf ( y)hy 12. 1 ( x)lxx P ( x)( y)lxy 13. 1 ( x)(dx (Ex F x)) P 2 ( x) F x P ( x) Dx 14. 1 ( x)(mx Lx) P 2 ( x)(nx Lx) P ( x)(nx Mx) 2

15. 1 ( x)( y)(f x Ay) P ( y)( x)(f x Ay) 16. 1 ( x)lx ( y)my P ( x)(lx ( y)my) 17. 1 ( x)( y)rxy P ( y)( x)ryx 18. ( y)(f y ( x)f x) 19. ( y)(( x)f x F y) 20. 1 ( x)(qx Rx) P 2 ( x)(qx Rx) P ( x)rx 21. ( x)f x ( x) F x 22. 1 ( x)(dx Ex) P Da (( y)(ey F y) F a) 23. 1 ( x)(lx Mx) P 2 ( x)(mx Nx) P ( x)lx ( y)ny 24. 1 ( x)jx ( y)ky P 2 ( x)(jx Kx) P ( y)ky 25. ( x)(f x Gx) (( x)f x ( x)gx) 26. ( x)(f x Gx) (( x)f x ( x)gx) 3

27. 1 ( z)( x)( y)szzxy P ( x)( y)( w)sxxyw 28. ( x)( y)(f xy (F xy F xy)) 29. ( x)( y)f xyy ( x)( y)( z)f yzx 30. ( x)( y)f xy ( y)( x)f xy 31. 1 ( x)(f x Gx) P 2 ( x)(f x Hx) P 3 ( y)(gy F y) ( x)(gx Hx) P ( y)(gy Hy) 32. ( x)(( y) Gxy ( z)gxz) 33. ( x)(f x ( y)gxy) ( x)( y)(f x Gxy) 34. 1 ( z)(gzz Hf z) P 2 ( x)(( y)f xy Gxx) P ( x)(( y)f xy Hfx) 35. 1 ( x)(qx Rx) P 2 ( x)(sx T x) P ( x)(rx Sx) ( y)(qy T y) 36. 1 ( x)(mx Lx) P 2 ( x)(nx Lx) P ( x)(nx Mx) 37. 1 ( x)(gx ( y)(hy Iy)) P ( x)gx ( y)(hy Iy) 4

38. 1 ( x)(sx (Sx T x)) P ( y) T y 39. 1 ( x)(f x Gx) ( ( y)gy ( z)f z) P 2 ( z)f z P ( x)(f x Gx) ( y)(gy Hy) 40. 1 ( x)(mx Lx) P 2 ( x)(ix Ax) P ( x)(lx Ix) ( x)(mx Ax) 41. 1 ( x)(nx Ox) P P a (( y)(p y Ny) Oa) 42. 1 ( x)ux ( y)((uy T y) W y) P 2 ( x)u x P 3 ( x)w x P ( x)(ux W x) 43. 1 ( x)jx ( y)(f y Gy) P ( x)(jx F x) ( y)(jy Gy) 44. 1 ( x)(sx (T x Ux)) P 2 ( x)(ux (Ex W x)) P ( x)(sx (T x Ex)) 45. 1 ( x)( y)(f xy Gyx) P ( y)( z)f zy ( y)( z) Gzy 5

46. 1 ( x)ax ( y)(by Cy) P 2 ( x)dx ( y)by P ( x)(ax Dx) ( y)cy 47. 1 ( x)((lx Mx) (Hx Jx)) P 2 ( x)((hx Jx) (Lx Mx)) P ( x)(lx Hx) 48. 1 ( x)(rx Cxx) P 2 ( w)p w P 3 ( y)p y ( z)(czz Rz) P ( x)(cxx Rx) 49. ( x)(f x Gx) (( x)f x ( x)gx) 50. ( x)(f x P ) (( x)f x P ) 51. 1 ( x)(ax ( y)(by Rxy)) P 2 ( x)(ax ( y)(sy Rxy)) P ( x)(bx Sx) 52. 1 ( x)((ax Bx) Cx) P 2 ( x)(dx Bx) P 3 ( x)(ax Dx) P ( x)(cx Dx) 53. 1 ( x)(hx (Bx T x)) P 2 ( x)(hx Bx) P 3 ( x)(hx T x) P ( x)(hx T x) 6

2 Answers to Problem Set #3, PD Exercises 1. 1 ( x)qx P 2 ( x)w x P ( x)(qx W x) 3 Qa 1 E 4 W a 2 E 5 Qa W a 3, 4 I 6 ( x)(qx W x) 5 V I 2. 1 ( x)(ax (Bx Cx)) P 2 Bd P Ad 3 Ad A 4 Ad (Bd Cd) 1 E 5 Bd Cd 3, 4 E 6 Bd 5 E 7 Bd 2 R 8 Ad 3 7 I 3. 1 ( x)jxx P 2 ( y) Ry P ( y)(jyy Ry) 3 Jaa 1 E 4 Ra 2 E 5 Jaa Ra 3, 4 I 6 ( y)(jyy Ry) 5 V I 7

4. 1 ( x)(lx (Mx Bx)) P 2 ( y)(my Jy) P ( x)(lx Jx) 3 Lc A 4 Lc (Mc Bc) 1 E 5 Mc Bc 3, 4 E 6 Mc 5 E 7 Mc Jc 2 E 8 Jc 6, 7 E 9 Lc Jc 3 8 I 10 ( x)(lx Jx) 9 V I 5. 1 ( w)( z)(bzw Kzw) P 2 Bcd P Kcd 3 Kcd A 4 ( z)(bzd Kzd) 1 E 5 Bcd Kcd 4 E 6 Bcd 3, 5 E 7 Bcd 2 R 8 Kcd 3 7 I 6. 1 ( x)(ax Bx) P ( x)(bx Cx) (Ak Ck) 2 ( x)(bx Cx) A 3 Ak A 4 Ak Bk 1 E 5 Bk 3, 4 E 6 Bk Ck 2 E 7 Ck 5, 6 E 8 Ak Ck 3 7 I 9 ( x)(bx Cx) (Ak Ck) 2 8 I 8

7. 1 ( x)jx ( y)ky P ( x)(jx ( y)ky) 2 Ja A 3 ( x)jx 2 I 4 ( y)ky 1, 3 E 5 Ja ( y)ky 2 4 I 6 ( x)(jx ( y)ky) 5 I 8. 1 ( x)bx Be P ( x)bx Be 2 ( x)bx A 3 Be 1, 2 E 4 Be A 5 ( x)bx 4 I 6 ( x)bx Be 2 3, 4 5 I 9. 1 ( x)(p x W x) P 2 P a P W a 3 W a A 4 P a W a 2, 3 I 5 ( x)(p x W x) 4 I 6 ( x)(p x W x) 1 R 7 W a 3 6 E 10. 1 ( x)(f x Gx) P ( y)f y 2 F a Ga A 3 F a 2 E 4 ( y)f y 3 I 5 ( y)f y 1, 2 4 E 9

11. 1 ( z)((f z Gzf) Hz) P ( x)gxf ( y)hy 2 (F k Gkf) Hk A 3 F k Gkf 2 E 4 Gkf 3 E 5 ( x)gxf 4 I 6 Hk 2 E 7 ( y)hy 6 I 8 ( x)gxf ( y)hy 5, 7 I 9 ( x)gxf ( y)hy 1, 2 8 E 12. 1 ( x)lxx P ( x)( y)lxy 2 Laa A 3 ( y)lay 2 I 4 ( x)( y)lxy 3 I 5 ( x)( y)lxy 1, 2 4 E 13. 1 ( x)(dx (Ex F x)) P 2 ( x) F x P ( x) Dx 3 F a A 4 Da (Ea F a) 1 E 5 Da A 6 Ea F a 4, 5 E 7 F a 6 E 8 F a 3 R 9 Da 5 8 I 10 ( x) Dx 9 I 11 ( x) Dx 2, 3 10 E 10

14. 1 ( x)(mx Lx) P 2 ( x)(nx Lx) P ( x)(nx Mx) 3 Na La A 4 Ma A 5 Ma La 1 E 6 La 4, 5 E 7 La 3 E 8 Ma 4 7 I 9 Na 3 E 10 Na Ma 9, 8 I 11 ( x)(nx Mx) 10 I 12 ( x)(nx Mx) 2, 3 11 E 15. 1 ( x)( y)(f x Ay) P ( y)( x)(f x Ay) 2 ( y)(f a Ay) A 3 F a Ab 2 E 4 ( x)(f x Ab) 3 I 5 ( y)( x)(f x Ay) 4 V I 6 ( y)( x)(f x Ay) 1, 2 5 E 16. 1 ( x)lx ( y)my P ( x)(lx ( y)my) 2 La A 3 ( x)lx 2 I 4 ( y)my 1, 3 E 5 La ( y)my 2 4 I 6 ( x)(lx ( y)my) 5 V I 11

17. 1 ( x)( y)rxy P ( y)( x)ryx 2 ( y)ray A 3 Rab A 4 ( x)rax 3 I 5 ( y)( x)ryx 4 I 6 ( y)( x)ryx 2, 3 5 E 7 ( y)( x)ryx 1, 2 6 E 18. ( y)(f y ( x)f x) 1 F a A 2 ( x)f x 1 I 3 F a ( x)f x 1 2 I 4 ( y)(f y ( x)f x) 3 V I 19. ( y)(( x)f x F y) 1 ( x)f x A 2 F a 1 E 3 ( x)f x F a 1 2 I 4 ( y)(( x)f x F y) 3 V I 20. 1 ( x)(qx Rx) P 2 ( x)(qx Rx) P ( x)rx 3 Qa Ra A 4 Qa A 5 Qa Ra 1 E 6 Ra 4, 5 E 7 Ra A 8 Ra 7 R 9 Ra 3, 4 6, 7 8 E 10 ( x)rx 9 I 11 ( x)rx 2, 3 10 E 12

21. ( x)f x ( x) F x 1 ( x)f x A 2 ( x) F x A 3 F a A 4 ( x) F x 3 I 5 ( x) F x 2 R 6 F a 3 5 E 7 ( x)f x 6 V I 8 ( x)f x 1 R 9 ( x) F x 2 8 E 10 ( x)f x ( x) F x 1 9 I 22. 1 ( x)(dx Ex) P Da (( y)(ey F y) F a) 2 Da A 3 ( y)(ey F y) A 4 Da Ea 1 E 5 Ea 2, 4 E 6 Ea F a 3 E 7 F a 5, 6 E 8 ( y)(ey F y) F a 3 7 I 9 Da (( y)(ey F y) F a) 2 8 I 23. 1 ( x)(lx Mx) P 2 ( x)(mx Nx) P ( x)lx ( y)ny 3 ( x)lx A 4 Ld A 5 Ld Md 1 E 6 Md Nd 2 E 7 Md 4, 5 E 8 Nd 6, 7 E 9 ( y)ny 8 I 10 ( y)ny 3, 4 9 E 11 ( x)lx ( y)ny 3 10 I 13

24. 1 ( x)jx ( y)ky P 2 ( x)(jx Kx) P ( y)ky 3 ( x)jx A 4 Ja A 5 Ja Ka 2 E 6 Ka 4, 5 E 7 ( y)ky 6 I 8 ( y)ky 3, 4 7 E 9 ( y)ky A 10 ( y)ky 9 R 11 ( y)ky 1, 3 8, 9 10 E 25. ( x)(f x Gx) (( x)f x ( x)gx) 1 ( x)(f x Gx) A 2 ( x)f x A 3 F a A 4 F a Ga 1 E 5 Ga 3, 4 E 6 ( x)gx 5 I 7 ( x)gx 2, 3 6 E 8 ( x)f x ( x)gx 2 7 I 9 ( x)(f x Gx) (( x)f x ( x)gx) 1 8 I 26. ( x)(f x Gx) (( x)f x ( x)gx) 1 ( x)(f x Gx) A 2 F a Ga A 3 F a 2 E 4 ( x)f x 3 I 5 Ga 2 E 6 ( x)gx 5 I 7 ( x)f x ( x)gx 4, 6 I 8 ( x)f x ( x)gx 1, 2 7 E 9 ( x)(f x Gx) (( x)f x ( x)gx) 1 8 I 14

27. 1 ( z)( x)( y)szzxy P ( x)( y)( w)sxxyw 2 ( x)( y)saaxy 1 E 3 ( y)saaby 2 E 4 Saabc 3 E 5 ( w)saabw 4 I 6 ( y)( w)saayw 5 V I 7 ( x)( y)( w)sxxyw 6 V I 28. ( x)( y)(f xy (F xy F xy)) 1 F ab A 2 F ab F ab 1 E 3 F ab (F ab F ab) 1 2 I 4 ( y)(f ay (F ay F ay)) 3 V I 5 ( x)( y)(f xy (F xy F xy)) 4 V I 29. ( x)( y)f xyy ( x)( y)( z)f yzx 1 ( x)( y)f xyy A 2 ( y)f syy 1 E 3 F stt 2 E 4 ( z)f szt 3 I 5 ( y)( z)f yzt 4 V I 6 ( x)( y)( z)f yzx 5 V I 7 ( x)( y)f xyy ( x)( y)( z)f yzx 1 6 I 30. ( x)( y)f xy ( y)( x)f xy 1 ( x)( y)f xy A 2 ( y)f ay A 3 F ab 2 E 4 ( x)f xb 3 I 5 ( y)( x)f xy 4 V I 6 ( y)( x)f xy 1, 2 5 E 7 ( x)( y)f xy ( y)( x)f xy 1 6 I 15

31. 1 ( x)(f x Gx) P 2 ( x)(f x Hx) P 3 ( y)(gy F y) ( x)(gx Hx) P ( y)(gy Hy) 4 F a Ga A 5 F a 4 E 6 Ga 4 E 7 Ga F a 6, 5 I 8 ( y)(gy F y) 7 I 9 ( x)(gx Hx) 3, 8 E 10 Gb Hb A 11 ( y)(gy Hy) 10 I 12 ( y)(gy Hy) 9, 10 11 E 13 ( y)(gy Hy) 1, 4 12 E 32. ( x)(( y) Gxy ( z)gxz) 1 ( y) Gay A 2 Gab A 3 ( z)gaz A 4 Gab 3 E 5 Gab 2 R 6 ( z)gaz 3 5 I 7 ( z)gaz 1, 2 6 E 8 ( y) Gay ( z)gaz 1 7 I 9 ( x)(( y) Gxy ( z)gxz) 8 V I 16

33. ( x)(f x ( y)gxy) ( x)( y)(f x Gxy) 1 ( x)(f x ( y)gxy) A 2 F a A 3 F a ( y)gay 1 E 4 ( y)gay 2, 3 E 5 Gab 4 E 6 F a Gab 2 5 I 7 ( y)(f a Gay) 6 V I 8 ( x)( y)(f x Gxy) 7 V I 9 ( x)(f x ( y)gxy) ( x)( y)(f x Gxy) 1 8 I 34. 1 ( z)(gzz Hf z) P 2 ( x)(( y)f xy Gxx) P ( x)(( y)f xy Hfx) 3 ( y)f ky Gkk A 4 Gkk Hfk 1 E 5 Gkk 3 E 6 Hfk 4, 5 E 7 ( y)f ky 3 E 8 ( y)f ky Hfk 7, 6 I 9 ( x)(( y)f xy Hfx) 8 I 10 ( x)(( y)f xy Hfx) 2, 3 9 E 17

35. 1 ( x)(qx Rx) P 2 ( x)(sx T x) P ( x)(rx Sx) ( y)(qy T y) 3 ( x)(rx Sx) A 4 Qa A 5 Qa Ra 1 E 6 Ra 4, 5 E 7 Ra Sa 3 E 8 Sa 6, 7 E 9 Sa T a 2 E 10 T a 8, 9 E 11 Qa T a 4 10 I 12 ( y)(qy T y) 11 V I 13 ( x)(rx Sx) ( y)(qy T y) 3 12 I 36. 1 ( x)(mx Lx) P 2 ( x)(nx Lx) P ( x)(nx Mx) 3 Na La A 4 Ma A 5 Ma La 1 E 6 La 4, 5 E 7 La 3 E 8 Ma 4 7 I 9 Na 3 E 10 Na Ma 9, 8 I 11 ( x)(nx Mx) 10 I 12 ( x)(nx Mx) 2, 3 11 E 18

37. 1 ( x)(gx ( y)(hy Iy)) P ( x)gx ( y)(hy Iy) 2 ( x)gx A 3 Ha A 4 Ga ( y)(hy Iy) 1 E 5 Ga 2 E 6 ( y)(hy Iy) 4, 5 E 7 Ha Ia 6 E 8 Ia 3, 7 E 9 Ha Ia 3 8 I 10 ( y)(hy Iy) 9 V I 11 ( x)gx ( y)(hy Iy) 2 10 I 38. 1 ( x)(sx (Sx T x)) P ( y) T y 2 Sa (Sa T a) A 3 Sa 2 E 4 (Sa T a) 2 E 5 T a A 6 Sa T a 3, 5 I 7 (Sa T a) 4 R 8 T a 5 7 I 9 ( y) T y 8 I 10 ( y) T y 1, 2 9 E 39. 1 ( x)(f x Gx) ( ( y)gy ( z)f z) P 2 ( z)f z P ( x)(f x Gx) ( y)(gy Hy) 3 ( x)(f x Gx) A 4 F a A 5 F a Ga 3 E 6 Ga 4, 5 E 7 Ga Ha 6 I 8 ( y)(gy Hy) 7 I 9 ( y)(gy Hy) 2, 4 8 E 10 ( x)(f x Gx) ( y)(gy Hy) 3 9 I 19

40. 1 ( x)(mx Lx) P 2 ( x)(ix Ax) P ( x)(lx Ix) ( x)(mx Ax) 3 ( x)(lx Ix) A 4 Ma A 5 Ma La 1 E 6 La 4, 5 E 7 La Ia 3 E 8 Ia 6, 7 E 9 Ia Aa 2 E 10 Aa 8, 9 E 11 Ma Aa 4 10 I 12 ( x)(mx Ax) 11 V I 13 ( x)(lx Ix) ( x)(mx Ax) 3 12 I 41. 1 ( x)(nx Ox) P P a (( y)(p y Ny) Oa) 2 P a A 3 ( y)(p y Ny) A 4 P a Na 3 E 5 Na 2, 4 E 6 Na Oa 1 E 7 Oa 5, 6 E 8 ( y)(p y Ny) Oa 3 7 I 9 P a (( y)(p y Ny) Oa) 2 8 I 10 ( x)(p x (( y)(p y Ny) Ox)) 9 V I 20

42. 1 ( x)ux ( y)((uy T y) W y) P 2 ( x)u x P 3 ( x)w x P ( x)(ux W x) 4 ( y)((uy T y) W y) 1, 2 E 5 Ua A 6 W b A 7 (Ua T a) W a 4 E 8 Ua T a 5 E 9 W a 7, 8 E 10 W a 3, 6 9 E 11 Ua W a 5, 10 I 12 ( x)(ux W x) 11 I 13 ( x)(ux W x) 2, 5 12 E 43. 1 ( x)jx ( y)(f y Gy) P ( x)(jx F x) ( y)(jy Gy) 2 ( x)(jx F x) A 3 Ja F a A 4 Ja 3 E 5 ( x)jx 4 I 6 ( y)(f y Gy) 1, 5 E 7 F a 3 E 8 F a Ga 6 E 9 Ga 7, 8 E 10 Ja Ga 4, 9 I 11 ( y)(jy Gy) 10 I 12 ( y)(jy Gy) 2, 3 11 E 13 ( x)(jx F x) ( y)(jy Gy) 2 12 I 21

44. 1 ( x)(sx (T x Ux)) P 2 ( x)(ux (Ex W x)) P ( x)(sx (T x Ex)) 3 Sa A 4 T a A 5 Sa (T a Ua) 1 E 6 T a Ua 3, 5 E 7 Ua 4, 6 E 8 Ua (Ea W a) 2 E 9 Ea W a 7, 8 E 10 Ea 9 E 11 T a Ea 4 10 I 12 Sa (T a Ea) 3 11 I 13 ( x)(sx (T x Ex)) 12 V I 45. 1 ( x)( y)(f xy Gyx) P ( y)( z)f zy ( y)( z) Gzy 2 ( y)(f by Gyb) 1 E 3 F bc Gcb 2 E 4 F bc 3 E 5 ( z)f zc 4 V I 6 ( y)( z)f zy 5 V I 7 Gcb 3 E 8 ( z) Gzb 7 V I 9 ( y)( z) Gzy 8 V I 10 ( y)( z)f zy ( y)( z) Gzy 6, 9 I 22

46. 1 ( x)ax ( y)(by Cy) P 2 ( x)dx ( y)by P ( x)(ax Dx) ( y)cy 3 ( x)(ax Dx) A 4 Ac Dc A 5 Dc 4 E 6 ( x)dx 5 I 7 ( y)by 2, 6 E 8 Bd A 9 Ac 4 E 10 ( x)ax 9 I 11 ( y)(by Cy) 1, 10 E 12 Bd Cd 11 E 13 Cd 8, 12 E 14 ( y)cy 13 I 15 ( y)cy 7, 8 14 E 16 ( y)cy 3, 4 15 E 17 ( x)(ax Dx) ( y)cy 3 16 I 47. 1 ( x)((lx Mx) (Hx Jx)) P 2 ( x)((hx Jx) (Lx Mx)) P ( x)(lx Hx) 3 La A 4 (La Ma) (Ha Ja) 1 E 5 La Ma 3 E 6 Ha Ja 4, 5 E 7 Ha 6 E 8 Ha A 9 (Ha Ja) (La Ma) 2 E 10 Ha Ja 8 E 11 La Ma 9, 10 E 12 La 11 E 13 La Ha 3 7, 8 12 I 14 ( x)(lx Hx) 13 V I 23

48. 1 ( x)(rx Cxx) P 2 ( w)p w P 3 ( y)p y ( z)(czz Rz) P ( x)(cxx Rx) 4 Ra Caa A 5 Ra A 6 Caa 4, 5 E 7 Caa A 8 P c A 9 ( y)p y 8 I 10 ( y)p y 2, 8 9 E 11 ( z)(czz Rz) 3, 10 E 12 Caa Ra 11 E 13 Ra 7, 12 E 14 Caa Ra 5 6, 7 13 I 15 ( x)(cxx Rx) 14 I 16 ( x)(cxx Rx) 1, 4 15 E 49. ( x)(f x Gx) (( x)f x ( x)gx) 1 ( x)(f x Gx) A 2 F a Ga 1 E 3 F a 2 E 4 ( x)f x 3 V I 5 Ga 2 E 6 ( x)gx 5 V I 7 ( x)f x ( x)gx 4, 6 I 8 ( x)f x ( x)gx A 9 ( x)f x 8 E 10 F a 9 E 11 ( x)gx 8 E 12 Ga 11 E 13 F a Ga 10, 12 I 14 ( x)(f x Gx) 13 V I 15 ( x)(f x Gx) (( x)f x ( x)gx) 1 7, 8 14 I 24

50. ( x)(f x P ) (( x)f x P ) 1 ( x)(f x P ) A 2 ( x)f x A 3 F a A 4 F a P 1 E 5 P 3, 4 E 6 P 2, 3 5 E 7 ( x)f x P 2 6 I 8 ( x)f x P A 9 F a A 10 ( x)f x 9 I 11 P 8, 10 E 12 F a P 9 11 I 13 ( x)(f x P ) 12 V I 14 ( x)(f x P ) (( x)f x P ) 1 7, 8 13 I 25

51. 1 ( x)(ax ( y)(by Rxy)) P 2 ( x)(ax ( y)(sy Rxy)) P ( x)(bx Sx) 3 Ba A 4 Ab ( y)(by Rby) A 5 Ab 4 E 6 ( y)(by Rby) 4 E 7 Ba Rba 6 E 8 Rba 3, 7 E 9 Ab ( y)(sy Rby) 2 E 10 ( y)(sy Rby) 5, 9 E 11 Sa A 12 Sa Rba 10 E 13 Rba 11, 12 E 14 Rba 8 R 15 Sa 11 14 I 16 Sa 1, 4 15 E 17 Ba Sa 3 16 I 18 ( x)(bx Sx) 17 V I 52. 1 ( x)((ax Bx) Cx) P 2 ( x)(dx Bx) P 3 ( x)(ax Dx) P ( x)(cx Dx) 4 Aa Da A 5 (Aa Ba) Ca 1 E 6 Da Ba 2 E 7 Aa 4 E 8 Da 4 E 9 Ba 6, 8 E 10 Aa Ba 7, 9 I 11 Ca 5, 10 E 12 Ca Da 11, 8 I 13 ( x)(cx Dx) 12 I 14 ( x)(cx Dx) 3, 4 13 E 26

53. 1 ( x)(hx (Bx T x)) P 2 ( x)(hx Bx) P 3 ( x)(hx T x) P ( x)(hx T x) 4 Ha Ba A 5 Ha 4 E 6 Ha (Ba T a) 1 E 7 Ba T a 5, 6 E 8 T a A 9 Ba 7, 8 E 10 Ba 4 E 11 T a 8 10 E 12 Ha T a 5, 11 I 13 ( x)(hx T x) 12 I 14 ( x)(hx T x) 2, 4 13 E 27

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