let x1, x2, x3, x4, x5, x6, x7 be non pair set ; :: thesis: for s being State of (STC0Circ (x1,x2,x3,x4,x5,x6,x7))
for a1, a2, a3, a4, a5, a6, a7 being Element of BOOLEAN st a1 = s . x1 & a2 = s . x2 & a3 = s . x3 & a4 = s . x4 & a5 = s . x5 & a6 = s . x6 & a7 = s . x7 holds
(Following (s,5)) . [<*(GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7))*>,xor2] = (((a1 '&' a2) 'or' (a2 '&' a3)) 'or' (a3 '&' a1)) 'xor' (((a5 '&' a6) 'or' (a6 '&' a7)) 'or' (a7 '&' a5))
set C = STC0Circ (x1,x2,x3,x4,x5,x6,x7);
set C1out = GFA0CarryOutput (x1,x2,x3);
set C2out = GFA0CarryOutput (x5,x6,x7);
set C1C2x = [<*(GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7))*>,xor2];
let s be State of (STC0Circ (x1,x2,x3,x4,x5,x6,x7)); :: thesis: for a1, a2, a3, a4, a5, a6, a7 being Element of BOOLEAN st a1 = s . x1 & a2 = s . x2 & a3 = s . x3 & a4 = s . x4 & a5 = s . x5 & a6 = s . x6 & a7 = s . x7 holds
(Following (s,5)) . [<*(GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7))*>,xor2] = (((a1 '&' a2) 'or' (a2 '&' a3)) 'or' (a3 '&' a1)) 'xor' (((a5 '&' a6) 'or' (a6 '&' a7)) 'or' (a7 '&' a5))
let a1, a2, a3, a4, a5, a6, a7 be Element of BOOLEAN ; :: thesis: ( a1 = s . x1 & a2 = s . x2 & a3 = s . x3 & a4 = s . x4 & a5 = s . x5 & a6 = s . x6 & a7 = s . x7 implies (Following (s,5)) . [<*(GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7))*>,xor2] = (((a1 '&' a2) 'or' (a2 '&' a3)) 'or' (a3 '&' a1)) 'xor' (((a5 '&' a6) 'or' (a6 '&' a7)) 'or' (a7 '&' a5)) )
assume A2: ( a1 = s . x1 & a2 = s . x2 & a3 = s . x3 & a4 = s . x4 & a5 = s . x5 & a6 = s . x6 & a7 = s . x7 ) ; :: thesis: (Following (s,5)) . [<*(GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7))*>,xor2] = (((a1 '&' a2) 'or' (a2 '&' a3)) 'or' (a3 '&' a1)) 'xor' (((a5 '&' a6) 'or' (a6 '&' a7)) 'or' (a7 '&' a5))
A5: ( (Following (s,4)) . (GFA0CarryOutput (x1,x2,x3)) = ((a1 '&' a2) 'or' (a2 '&' a3)) 'or' (a3 '&' a1) & (Following (s,4)) . (GFA0CarryOutput (x5,x6,x7)) = ((a5 '&' a6) 'or' (a6 '&' a7)) 'or' (a7 '&' a5) ) by A2, LmSTC0S15s4a;
Following (s,(4 + 1)) = Following (Following (s,4)) by FACIRC_1:12;
hence (Following (s,5)) . [<*(GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7))*>,xor2] = (((a1 '&' a2) 'or' (a2 '&' a3)) 'or' (a3 '&' a1)) 'xor' (((a5 '&' a6) 'or' (a6 '&' a7)) 'or' (a7 '&' a5)) by A5, LmSTC0S15S1s4; :: thesis: verum