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