let x, y, c be non pair set ; :: thesis: for s being State of (BorrowCirc (x,y,c))
for a1, a2, a3 being Element of BOOLEAN st a1 = s . x & a2 = s . y & a3 = s . c holds
( (Following (s,2)) . (BorrowOutput (x,y,c)) = ((('not' a1) '&' a2) 'or' (a2 '&' a3)) 'or' (('not' a1) '&' a3) & (Following (s,2)) . [<*x,y*>,and2a] = ('not' a1) '&' a2 & (Following (s,2)) . [<*y,c*>,and2] = a2 '&' a3 & (Following (s,2)) . [<*x,c*>,and2a] = ('not' a1) '&' a3 )
set S = BorrowStr (x,y,c);
reconsider x9 = x, y9 = y, c9 = c as Vertex of (BorrowStr (x,y,c)) by Th6;
let s be State of (BorrowCirc (x,y,c)); :: thesis: for a1, a2, a3 being Element of BOOLEAN st a1 = s . x & a2 = s . y & a3 = s . c holds
( (Following (s,2)) . (BorrowOutput (x,y,c)) = ((('not' a1) '&' a2) 'or' (a2 '&' a3)) 'or' (('not' a1) '&' a3) & (Following (s,2)) . [<*x,y*>,and2a] = ('not' a1) '&' a2 & (Following (s,2)) . [<*y,c*>,and2] = a2 '&' a3 & (Following (s,2)) . [<*x,c*>,and2a] = ('not' a1) '&' a3 )
set xy = <*x,y*>;
set yc = <*y,c*>;
set xc = <*x,c*>;
set xy1 = [<*x,y*>,and2a];
set yc1 = [<*y,c*>,and2];
set xc1 = [<*x,c*>,and2a];
A1: Following (s,2) = Following (Following s) by FACIRC_1:15;
let a1, a2, a3 be Element of BOOLEAN ; :: thesis: ( a1 = s . x & a2 = s . y & a3 = s . c implies ( (Following (s,2)) . (BorrowOutput (x,y,c)) = ((('not' a1) '&' a2) 'or' (a2 '&' a3)) 'or' (('not' a1) '&' a3) & (Following (s,2)) . [<*x,y*>,and2a] = ('not' a1) '&' a2 & (Following (s,2)) . [<*y,c*>,and2] = a2 '&' a3 & (Following (s,2)) . [<*x,c*>,and2a] = ('not' a1) '&' a3 ) )
assume A2: ( a1 = s . x & a2 = s . y & a3 = s . c ) ; :: thesis: ( (Following (s,2)) . (BorrowOutput (x,y,c)) = ((('not' a1) '&' a2) 'or' (a2 '&' a3)) 'or' (('not' a1) '&' a3) & (Following (s,2)) . [<*x,y*>,and2a] = ('not' a1) '&' a2 & (Following (s,2)) . [<*y,c*>,and2] = a2 '&' a3 & (Following (s,2)) . [<*x,c*>,and2a] = ('not' a1) '&' a3 )
A3: (Following s) . [<*x,c*>,and2a] = ('not' a1) '&' a3 by A2, Lm1;
( (Following s) . [<*x,y*>,and2a] = ('not' a1) '&' a2 & (Following s) . [<*y,c*>,and2] = a2 '&' a3 ) by A2, Lm1;
hence (Following (s,2)) . (BorrowOutput (x,y,c)) = ((('not' a1) '&' a2) 'or' (a2 '&' a3)) 'or' (('not' a1) '&' a3) by A1, A3, Th13; :: thesis: ( (Following (s,2)) . [<*x,y*>,and2a] = ('not' a1) '&' a2 & (Following (s,2)) . [<*y,c*>,and2] = a2 '&' a3 & (Following (s,2)) . [<*x,c*>,and2a] = ('not' a1) '&' a3 )
y in InputVertices (BorrowStr (x,y,c)) by Th8;
then A4: (Following s) . y9 = s . y by CIRCUIT2:def 5;
c in InputVertices (BorrowStr (x,y,c)) by Th8;
then A5: (Following s) . c9 = s . c by CIRCUIT2:def 5;
x in InputVertices (BorrowStr (x,y,c)) by Th8;
then (Following s) . x9 = s . x by CIRCUIT2:def 5;
hence ( (Following (s,2)) . [<*x,y*>,and2a] = ('not' a1) '&' a2 & (Following (s,2)) . [<*y,c*>,and2] = a2 '&' a3 & (Following (s,2)) . [<*x,c*>,and2a] = ('not' a1) '&' a3 ) by A2, A4, A5, A1, Lm1; :: thesis: verum