let x, y, c be non pair set ; :: thesis: for s being State of (BitSubtracterWithBorrowCirc x,y,c) holds Following s,2 is stable
set S = BitSubtracterWithBorrowStr x,y,c;
set S1 = 2GatesCircStr x,y,c,'xor' ;
set S2 = BorrowStr x,y,c;
set A = BitSubtracterWithBorrowCirc x,y,c;
set A1 = BitSubtracterCirc x,y,c;
set A2 = BorrowCirc x,y,c;
let s be State of (BitSubtracterWithBorrowCirc x,y,c); :: thesis: Following s,2 is stable
reconsider s1 = s | the carrier of (2GatesCircStr x,y,c,'xor' ) as State of (BitSubtracterCirc x,y,c) by FACIRC_1:26;
reconsider s2 = s | the carrier of (BorrowStr x,y,c) as State of (BorrowCirc x,y,c) by FACIRC_1:26;
reconsider t = s as State of ((BitSubtracterCirc x,y,c) +* (BorrowCirc x,y,c)) ;
A1: dom (Following s,3) = the carrier of (BitSubtracterWithBorrowStr x,y,c) by CIRCUIT1:4;
A2: the carrier of (BitSubtracterWithBorrowStr x,y,c) = the carrier of (2GatesCircStr x,y,c,'xor' ) \/ the carrier of (BorrowStr x,y,c) by CIRCCOMB:def 2;
not InputVertices (2GatesCircStr x,y,c,'xor' ) is with_pair by FACIRC_1:59;
then InnerVertices (BorrowStr x,y,c) misses InputVertices (2GatesCircStr x,y,c,'xor' ) by Th1, FACIRC_1:5;
then A3: ( Following s1,2 = (Following t,2) | the carrier of (2GatesCircStr x,y,c,'xor' ) & Following s1,3 = (Following t,3) | the carrier of (2GatesCircStr x,y,c,'xor' ) ) by FACIRC_1:30;
Following s1,2 is stable by FACIRC_1:63;
then A4: Following s1,2 = Following (Following s1,2) by CIRCUIT2:def 6
.= Following s1,(2 + 1) by FACIRC_1:12 ;
not InputVertices (BorrowStr x,y,c) is with_pair by Th2;
then InnerVertices (2GatesCircStr x,y,c,'xor' ) misses InputVertices (BorrowStr x,y,c) by FACIRC_1:5, FACIRC_1:58;
then A5: ( Following s2,2 = (Following t,2) | the carrier of (BorrowStr x,y,c) & Following s2,3 = (Following t,3) | the carrier of (BorrowStr x,y,c) ) by FACIRC_1:31;
Following s2,2 is stable by Th18;
then A6: Following s2,2 = Following (Following s2,2) by CIRCUIT2:def 6
.= Following s2,(2 + 1) by FACIRC_1:12 ;
A7: ( dom (Following s1,2) = the carrier of (2GatesCircStr x,y,c,'xor' ) & dom (Following s2,2) = the carrier of (BorrowStr x,y,c) ) by CIRCUIT1:4;

A8: now

let a be set ; :: thesis: ( a in the carrier of (BitSubtracterWithBorrowStr x,y,c) implies (Following s,2) . a = (Following (Following s,2)) . a )
assume a in the carrier of (BitSubtracterWithBorrowStr x,y,c) ; :: thesis: (Following s,2) . a = (Following (Following s,2)) . a
then ( a in the carrier of (2GatesCircStr x,y,c,'xor' ) or a in the carrier of (BorrowStr x,y,c) ) by A2, XBOOLE_0:def 3;
then ( ( (Following s,2) . a = (Following s1,2) . a & (Following s,3) . a = (Following s1,3) . a ) or ( (Following s,2) . a = (Following s2,2) . a & (Following s,3) . a = (Following s2,3) . a ) ) by A3, A5, A4, A6, A7, FUNCT_1:70;
hence (Following s,2) . a = (Following (Following s,2)) . a by A4, A6, FACIRC_1:12; :: thesis: verum

end;

( Following s,(2 + 1) = Following (Following s,2) & dom (Following s,2) = the carrier of (BitSubtracterWithBorrowStr x,y,c) ) by CIRCUIT1:4, FACIRC_1:12;
hence Following s,2 = Following (Following s,2) by A1, A8, FUNCT_1:9; :: according to CIRCUIT2:def 6 :: thesis: verum