let g0, g1, g2, g3, g4, g5, g6, g7, g8, g9, g10, g11, g12, a0, a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, z, p be set ; :: thesis:
assume that
A1: not g0 is empty and
A2: z is empty and
A3: ( not b0 is empty iff not XOR2 (p,a11) is empty ) and
A4: ( not b1 is empty iff not XOR2 (a0,(AND2 (g1,b0))) is empty ) and
A5: ( not b2 is empty iff not XOR2 (a1,(AND2 (g2,b0))) is empty ) and
A6: ( not b3 is empty iff not XOR2 (a2,(AND2 (g3,b0))) is empty ) and
A7: ( not b4 is empty iff not XOR2 (a3,(AND2 (g4,b0))) is empty ) and
A8: ( not b5 is empty iff not XOR2 (a4,(AND2 (g5,b0))) is empty ) and
A9: ( not b6 is empty iff not XOR2 (a5,(AND2 (g6,b0))) is empty ) and
A10: ( not b7 is empty iff not XOR2 (a6,(AND2 (g7,b0))) is empty ) and
A11: ( not b8 is empty iff not XOR2 (a7,(AND2 (g8,b0))) is empty ) and
A12: ( not b9 is empty iff not XOR2 (a8,(AND2 (g9,b0))) is empty ) and
A13: ( not b10 is empty iff not XOR2 (a9,(AND2 (g10,b0))) is empty ) and
A14: ( not b11 is empty iff not XOR2 (a10,(AND2 (g11,b0))) is empty ) ; :: thesis:
( ( ( not a10 is empty & ( g11 is empty or a11 is empty ) & ( g11 is empty or p is empty ) ) or ( a10 is empty & not g11 is empty & not a11 is empty & ( g11 is empty or p is empty ) ) or ( ( a10 is empty or ( not g11 is empty & not a11 is empty ) ) & ( not a10 is empty or g11 is empty or a11 is empty ) & not g11 is empty & not p is empty ) ) iff not XOR2 ((XOR2 (a10,(AND2 (g11,a11)))),(XOR2 (z,(AND2 (g11,p))))) is empty ) by A2;
hence ( not b11 is empty iff not XOR2 ((XOR2 (a10,(AND2 (g11,a11)))),(XOR2 (z,(AND2 (g11,p))))) is empty ) by A3, A14; :: thesis:
( ( ( not a9 is empty & ( g10 is empty or a11 is empty ) & ( g10 is empty or p is empty ) ) or ( a9 is empty & not g10 is empty & not a11 is empty & ( g10 is empty or p is empty ) ) or ( ( a9 is empty or ( not g10 is empty & not a11 is empty ) ) & ( not a9 is empty or g10 is empty or a11 is empty ) & not g10 is empty & not p is empty ) ) iff not XOR2 ((XOR2 (a9,(AND2 (g10,a11)))),(XOR2 (z,(AND2 (g10,p))))) is empty ) by A2;
hence ( not b10 is empty iff not XOR2 ((XOR2 (a9,(AND2 (g10,a11)))),(XOR2 (z,(AND2 (g10,p))))) is empty ) by A3, A13; :: thesis:
( ( ( not a8 is empty & ( g9 is empty or a11 is empty ) & ( g9 is empty or p is empty ) ) or ( a8 is empty & not g9 is empty & not a11 is empty & ( g9 is empty or p is empty ) ) or ( ( a8 is empty or ( not g9 is empty & not a11 is empty ) ) & ( not a8 is empty or g9 is empty or a11 is empty ) & not g9 is empty & not p is empty ) ) iff not XOR2 ((XOR2 (a8,(AND2 (g9,a11)))),(XOR2 (z,(AND2 (g9,p))))) is empty ) by A2;
hence ( not b9 is empty iff not XOR2 ((XOR2 (a8,(AND2 (g9,a11)))),(XOR2 (z,(AND2 (g9,p))))) is empty ) by A3, A12; :: thesis:
( ( ( not a7 is empty & ( g8 is empty or a11 is empty ) & ( g8 is empty or p is empty ) ) or ( a7 is empty & not g8 is empty & not a11 is empty & ( g8 is empty or p is empty ) ) or ( ( a7 is empty or ( not g8 is empty & not a11 is empty ) ) & ( not a7 is empty or g8 is empty or a11 is empty ) & not g8 is empty & not p is empty ) ) iff not XOR2 ((XOR2 (a7,(AND2 (g8,a11)))),(XOR2 (z,(AND2 (g8,p))))) is empty ) by A2;
hence ( not b8 is empty iff not XOR2 ((XOR2 (a7,(AND2 (g8,a11)))),(XOR2 (z,(AND2 (g8,p))))) is empty ) by A3, A11; :: thesis:
( ( ( not a6 is empty & ( g7 is empty or a11 is empty ) & ( g7 is empty or p is empty ) ) or ( a6 is empty & not g7 is empty & not a11 is empty & ( g7 is empty or p is empty ) ) or ( ( a6 is empty or ( not g7 is empty & not a11 is empty ) ) & ( not a6 is empty or g7 is empty or a11 is empty ) & not g7 is empty & not p is empty ) ) iff not XOR2 ((XOR2 (a6,(AND2 (g7,a11)))),(XOR2 (z,(AND2 (g7,p))))) is empty ) by A2;
hence ( not b7 is empty iff not XOR2 ((XOR2 (a6,(AND2 (g7,a11)))),(XOR2 (z,(AND2 (g7,p))))) is empty ) by A3, A10; :: thesis:
( ( ( not a5 is empty & ( g6 is empty or a11 is empty ) & ( g6 is empty or p is empty ) ) or ( a5 is empty & not g6 is empty & not a11 is empty & ( g6 is empty or p is empty ) ) or ( ( a5 is empty or ( not g6 is empty & not a11 is empty ) ) & ( not a5 is empty or g6 is empty or a11 is empty ) & not g6 is empty & not p is empty ) ) iff not XOR2 ((XOR2 (a5,(AND2 (g6,a11)))),(XOR2 (z,(AND2 (g6,p))))) is empty ) by A2;
hence ( not b6 is empty iff not XOR2 ((XOR2 (a5,(AND2 (g6,a11)))),(XOR2 (z,(AND2 (g6,p))))) is empty ) by A3, A9; :: thesis:
( ( ( not a4 is empty & ( g5 is empty or a11 is empty ) & ( g5 is empty or p is empty ) ) or ( a4 is empty & not g5 is empty & not a11 is empty & ( g5 is empty or p is empty ) ) or ( ( a4 is empty or ( not g5 is empty & not a11 is empty ) ) & ( not a4 is empty or g5 is empty or a11 is empty ) & not g5 is empty & not p is empty ) ) iff not XOR2 ((XOR2 (a4,(AND2 (g5,a11)))),(XOR2 (z,(AND2 (g5,p))))) is empty ) by A2;
hence ( not b5 is empty iff not XOR2 ((XOR2 (a4,(AND2 (g5,a11)))),(XOR2 (z,(AND2 (g5,p))))) is empty ) by A3, A8; :: thesis:
( ( ( not a3 is empty & ( g4 is empty or a11 is empty ) & ( g4 is empty or p is empty ) ) or ( a3 is empty & not g4 is empty & not a11 is empty & ( g4 is empty or p is empty ) ) or ( ( a3 is empty or ( not g4 is empty & not a11 is empty ) ) & ( not a3 is empty or g4 is empty or a11 is empty ) & not g4 is empty & not p is empty ) ) iff not XOR2 ((XOR2 (a3,(AND2 (g4,a11)))),(XOR2 (z,(AND2 (g4,p))))) is empty ) by A2;
hence ( not b4 is empty iff not XOR2 ((XOR2 (a3,(AND2 (g4,a11)))),(XOR2 (z,(AND2 (g4,p))))) is empty ) by A3, A7; :: thesis:
( ( ( not a2 is empty & ( g3 is empty or a11 is empty ) & ( g3 is empty or p is empty ) ) or ( a2 is empty & not g3 is empty & not a11 is empty & ( g3 is empty or p is empty ) ) or ( ( a2 is empty or ( not g3 is empty & not a11 is empty ) ) & ( not a2 is empty or g3 is empty or a11 is empty ) & not g3 is empty & not p is empty ) ) iff not XOR2 ((XOR2 (a2,(AND2 (g3,a11)))),(XOR2 (z,(AND2 (g3,p))))) is empty ) by A2;
hence ( not b3 is empty iff not XOR2 ((XOR2 (a2,(AND2 (g3,a11)))),(XOR2 (z,(AND2 (g3,p))))) is empty ) by A3, A6; :: thesis:
( ( ( not a1 is empty & ( g2 is empty or a11 is empty ) & ( g2 is empty or p is empty ) ) or ( a1 is empty & not g2 is empty & not a11 is empty & ( g2 is empty or p is empty ) ) or ( ( a1 is empty or ( not g2 is empty & not a11 is empty ) ) & ( not a1 is empty or g2 is empty or a11 is empty ) & not g2 is empty & not p is empty ) ) iff not XOR2 ((XOR2 (a1,(AND2 (g2,a11)))),(XOR2 (z,(AND2 (g2,p))))) is empty ) by A2;
hence ( not b2 is empty iff not XOR2 ((XOR2 (a1,(AND2 (g2,a11)))),(XOR2 (z,(AND2 (g2,p))))) is empty ) by A3, A5; :: thesis:
( ( ( not a0 is empty & ( g1 is empty or a11 is empty ) & ( g1 is empty or p is empty ) ) or ( a0 is empty & not g1 is empty & not a11 is empty & ( g1 is empty or p is empty ) ) or ( ( a0 is empty or ( not g1 is empty & not a11 is empty ) ) & ( not a0 is empty or g1 is empty or a11 is empty ) & not g1 is empty & not p is empty ) ) iff not XOR2 ((XOR2 (a0,(AND2 (g1,a11)))),(XOR2 (z,(AND2 (g1,p))))) is empty ) by A2;
hence ( not b1 is empty iff not XOR2 ((XOR2 (a0,(AND2 (g1,a11)))),(XOR2 (z,(AND2 (g1,p))))) is empty ) by A3, A4; :: thesis:
( ( ( not z is empty & ( g0 is empty or a11 is empty ) & ( g0 is empty or p is empty ) ) or ( z is empty & not g0 is empty & not a11 is empty & ( g0 is empty or p is empty ) ) or ( ( z is empty or ( not g0 is empty & not a11 is empty ) ) & ( not z is empty or g0 is empty or a11 is empty ) & not g0 is empty & not p is empty ) ) iff not XOR2 ((XOR2 (z,(AND2 (g0,a11)))),(XOR2 (z,(AND2 (g0,p))))) is empty ) by A2;
hence ( not b0 is empty iff not XOR2 ((XOR2 (z,(AND2 (g0,a11)))),(XOR2 (z,(AND2 (g0,p))))) is empty ) by A1, A3; :: thesis: