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

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 A6: 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 (BorrowStr x,y,c) ;
A7: v in (InputVertices (BorrowStr x,y,c)) \/ (InnerVertices (BorrowStr x,y,c)) by A6, 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 A7, 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 A2, A5, Lm1, Th13; :: thesis: verum

end;

end;

end;

end;

hence Following s,2 = Following (Following s,2) by A1, FUNCT_1:9; :: according to CIRCUIT2:def 6 :: thesis: verum