let x, y, c be set ; :: thesis:
assume A1: ( x <> [<*y,c*>,and2 ] & y <> [<*x,c*>,and2a ] & 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 A2: ( a1 = s . x & a2 = s . y & 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 x' = x, y' = y, c' = c as Vertex of (BorrowStr x,y,c) by FSCIRC_1:6;
InputVertices (BorrowStr x,y,c) = {x,y,c} by A1, Th15;
then ( x in InputVertices (BorrowStr x,y,c) & y in InputVertices (BorrowStr x,y,c) & c in InputVertices (BorrowStr x,y,c) ) by ENUMSET1:def 1;
then A3: ( (Following s) . x' = s . x & (Following s) . y' = s . y & (Following s) . c' = s . c ) by CIRCUIT2:def 5;
A4: Following s,2 = Following (Following s) by FACIRC_1:15;
( (Following s) . [<*x,y*>,and2a ] = ('not' a1) '&' a2 & (Following s) . [<*y,c*>,and2 ] = a2 '&' a3 & (Following s) . [<*x,c*>,and2a ] = ('not' a1) '&' a3 ) by A2, Lm1;
hence (Following s,2) . (BorrowOutput x,y,c) = ((('not' a1) '&' a2) 'or' (a2 '&' a3)) 'or' (('not' a1) '&' a3) by A4, 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 A2, A3, A4, Lm1; :: thesis: