let x, y, c be set ; :: thesis:
assume that
A1: x <> [<*y,c*>,and2] and
A2: y <> [<*x,c*>,and2a] and
A3: c <> [<*x,y*>,and2a] ; :: thesis:
let s be State of (BorrowCirc (x,y,c)); :: thesis:
let a1, a2, a3 be Element of BOOLEAN ; :: thesis:
assume that
A4: a1 = s . x and
A5: a2 = s . y and
A6: a3 = s . c ; :: thesis:
set xy = [<*x,y*>,and2a];
set yc = [<*y,c*>,and2];
set cx = [<*x,c*>,and2a];
set S = BorrowStr (x,y,c);
reconsider x9 = x, y9 = y, c9 = c as Vertex of (BorrowStr (x,y,c)) by FSCIRC_1:6;
A7: InputVertices (BorrowStr (x,y,c)) = {x,y,c} by A1, A2, A3, Th15;
then A8: x in InputVertices (BorrowStr (x,y,c)) by ENUMSET1:def 1;
A9: y in InputVertices (BorrowStr (x,y,c)) by A7, ENUMSET1:def 1;
A10: c in InputVertices (BorrowStr (x,y,c)) by A7, ENUMSET1:def 1;
A11: (Following s) . x9 = s . x by A8, CIRCUIT2:def 5;
A12: (Following s) . y9 = s . y by A9, CIRCUIT2:def 5;
A13: (Following s) . c9 = s . c by A10, CIRCUIT2:def 5;
A14: Following (s,2) = Following (Following s) by FACIRC_1:15;
A15: (Following s) . [<*x,y*>,and2a] = ('not' a1) '&' a2 by A4, A5, A6, Lm1;
A16: (Following s) . [<*y,c*>,and2] = a2 '&' a3 by A4, A5, A6, Lm1;
(Following s) . [<*x,c*>,and2a] = ('not' a1) '&' a3 by A4, A5, A6, Lm1;
hence (Following (s,2)) . (BorrowOutput (x,y,c)) = ((('not' a1) '&' a2) 'or' (a2 '&' a3)) 'or' (('not' a1) '&' a3) by A14, A15, A16, Th22; :: thesis:
thus ( (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 A4, A5, A6, A11, A12, A13, A14, Lm1; :: thesis: