let x, y, c be set ; :: thesis: for s being State of (BorrowCirc (x,y,c))
for a1, a2, a3 being Element of BOOLEAN st a1 = s . [<*x,y*>,and2a] & a2 = s . [<*y,c*>,and2] & a3 = s . [<*x,c*>,and2a] holds
(Following s) . (BorrowOutput (x,y,c)) = (a1 'or' a2) 'or' a3
let s be State of (BorrowCirc (x,y,c)); :: thesis: for a1, a2, a3 being Element of BOOLEAN st a1 = s . [<*x,y*>,and2a] & a2 = s . [<*y,c*>,and2] & a3 = s . [<*x,c*>,and2a] holds
(Following s) . (BorrowOutput (x,y,c)) = (a1 'or' a2) 'or' a3
let a1, a2, a3 be Element of BOOLEAN ; :: thesis: ( a1 = s . [<*x,y*>,and2a] & a2 = s . [<*y,c*>,and2] & a3 = s . [<*x,c*>,and2a] implies (Following s) . (BorrowOutput (x,y,c)) = (a1 'or' a2) 'or' a3 )
assume that
A1: a1 = s . [<*x,y*>,and2a] and
A2: a2 = s . [<*y,c*>,and2] and
A3: a3 = s . [<*x,c*>,and2a] ; :: thesis: (Following s) . (BorrowOutput (x,y,c)) = (a1 'or' a2) 'or' a3
set xy = [<*x,y*>,and2a];
set yc = [<*y,c*>,and2];
set cx = [<*x,c*>,and2a];
set S = BorrowStr (x,y,c);
A4: InnerVertices (BorrowStr (x,y,c)) = the carrier' of (BorrowStr (x,y,c)) by FACIRC_1:37;
A5: dom s = the carrier of (BorrowStr (x,y,c)) by CIRCUIT1:3;
reconsider xy = [<*x,y*>,and2a], yc = [<*y,c*>,and2], cx = [<*x,c*>,and2a] as Element of InnerVertices (BorrowStr (x,y,c)) by FSCIRC_1:7;
thus (Following s) . (BorrowOutput (x,y,c)) = or3 . (s * <*xy,yc,cx*>) by A4, FACIRC_1:35
.= or3 . <*a1,a2,a3*> by A1, A2, A3, A5, FINSEQ_2:126
.= (a1 'or' a2) 'or' a3 by FACIRC_1:def 7 ; :: thesis: verum