let x, y, c be non pair set ; :: thesis: for s being State of holds Following s,2 is stable
set S = BorrowStr x,y,c;
reconsider xx = x, yy = y, cc = c as Vertex of by Th6;
let s be State of ; :: thesis: Following s,2 is stable
set a1 = s . xx;
set a2 = s . yy;
set a3 = s . cc;
set ffs = Following s,2;
set fffs = Following (Following s,2);
A1: Following s,2 = Following (Following s) by FACIRC_1:15;
A2: y in InputVertices (BorrowStr x,y,c) by Th8;
then (Following s) . y = s . yy by CIRCUIT2:def 5;
then A3: (Following s,2) . y = s . yy by A1, A2, CIRCUIT2:def 5;
s . yy = s . y ;
then A4: (Following s,2) . [<*x,c*>,and2a ] = ('not' (s . xx)) '&' (s . cc) by Lm2;
A5: x in InputVertices (BorrowStr x,y,c) by Th8;
then (Following s) . x = s . xx by CIRCUIT2:def 5;
then A6: (Following s,2) . x = s . xx by A1, A5, CIRCUIT2:def 5;
s . xx = s . x ;
then A7: (Following s,2) . [<*y,c*>,and2 ] = (s . yy) '&' (s . cc) by Lm2;
A8: c in InputVertices (BorrowStr x,y,c) by Th8;
then (Following s) . c = s . cc by CIRCUIT2:def 5;
then A9: (Following s,2) . c = s . cc by A1, A8, CIRCUIT2:def 5;
s . cc = s . c ;
then A10: (Following s,2) . [<*x,y*>,and2a ] = ('not' (s . xx)) '&' (s . yy) by Lm2;
A11: (Following s,2) . (BorrowOutput x,y,c) = ((('not' (s . xx)) '&' (s . yy)) 'or' ((s . yy) '&' (s . cc))) 'or' (('not' (s . xx)) '&' (s . cc)) by Lm2;

A12: now

let a be set ; :: thesis: ( a in the carrier of (BorrowStr x,y,c) implies (Following s,2) . a = (Following (Following s,2)) . a )
assume A13: a in the carrier of (BorrowStr x,y,c) ; :: thesis: (Following s,2) . a = (Following (Following s,2)) . a
then reconsider v = a as Vertex of ;
A14: v in (InputVertices (BorrowStr x,y,c)) \/ (InnerVertices (BorrowStr x,y,c)) by A13, XBOOLE_1:45;
thus (Following s,2) . a = (Following (Following s,2)) . a :: thesis: verum

proof

per cases ( v in InputVertices (BorrowStr x,y,c) or v in InnerVertices (BorrowStr x,y,c) ) by A14, XBOOLE_0:def 3;

suppose v in InputVertices (BorrowStr x,y,c) ; :: thesis: (Following s,2) . a = (Following (Following s,2)) . a

hence (Following s,2) . a = (Following (Following s,2)) . a by CIRCUIT2:def 5; :: thesis: verum

end;

suppose v in InnerVertices (BorrowStr x,y,c) ; :: thesis: (Following s,2) . a = (Following (Following s,2)) . a

then v in {[<*x,y*>,and2a ],[<*y,c*>,and2 ],[<*x,c*>,and2a ]} \/ {(BorrowOutput x,y,c)} by Th9;
then ( v in {[<*x,y*>,and2a ],[<*y,c*>,and2 ],[<*x,c*>,and2a ]} or v in {(BorrowOutput x,y,c)} ) by XBOOLE_0:def 3;
then ( v = [<*x,y*>,and2a ] or v = [<*y,c*>,and2 ] or v = [<*x,c*>,and2a ] or v = BorrowOutput x,y,c ) by ENUMSET1:def 1, TARSKI:def 1;
hence (Following s,2) . a = (Following (Following s,2)) . a by A11, A10, A7, A4, A6, A3, A9, Lm1, Th13; :: thesis: verum

end;

end;

end;

end;

( dom (Following (Following s,2)) = the carrier of (BorrowStr x,y,c) & dom (Following s,2) = the carrier of (BorrowStr x,y,c) ) by CIRCUIT1:4;
hence Following s,2 = Following (Following s,2) by A12, FUNCT_1:9; :: according to CIRCUIT2:def 6 :: thesis: verum