set J = Stop SCM+FSA;
set G = Goto 0;
let a be Int-Location ; :: thesis: for I being Program of SCM+FSA holds
( (card I) + 4 in dom (if>0 (a,(I ';' (Goto 0)),(Stop SCM+FSA))) & (if>0 (a,(I ';' (Goto 0)),(Stop SCM+FSA))) . ((card I) + 4) = goto (0 + ((card I) + 4)) )
let I be Program of SCM+FSA; :: thesis: ( (card I) + 4 in dom (if>0 (a,(I ';' (Goto 0)),(Stop SCM+FSA))) & (if>0 (a,(I ';' (Goto 0)),(Stop SCM+FSA))) . ((card I) + 4) = goto (0 + ((card I) + 4)) )
set I1 = I ';' (Goto 0);
set i = a >0_goto ((card (Stop SCM+FSA)) + 3);
set c4 = (card I) + 4;
set Lc4 = (card I) + 4;
set Mi = (((Macro (a >0_goto ((card (Stop SCM+FSA)) + 3))) ';' (Stop SCM+FSA)) ';' (Goto ((card (I ';' (Goto 0))) + 1))) ';' I;
x: card (Stop SCM+FSA) = 1 by COMPOS_1:46;
A1: card ((Goto 0) ';' (Stop SCM+FSA)) = (card (Goto 0)) + (card (Stop SCM+FSA)) by SCMFSA6A:61
.= 1 + 1 by x, SCMFSA8A:29
.= 2 ;
A2: if>0 (a,(I ';' (Goto 0)),(Stop SCM+FSA)) = ((((a >0_goto ((card (Stop SCM+FSA)) + 3)) ';' (Stop SCM+FSA)) ';' (Goto ((card (I ';' (Goto 0))) + 1))) ';' (I ';' (Goto 0))) ';' (Stop SCM+FSA) by SCMFSA8B:def 2
.= (((((Macro (a >0_goto ((card (Stop SCM+FSA)) + 3))) ';' (Stop SCM+FSA)) ';' (Goto ((card (I ';' (Goto 0))) + 1))) ';' I) ';' (Goto 0)) ';' (Stop SCM+FSA) by SCMFSA6A:67
.= ((((Macro (a >0_goto ((card (Stop SCM+FSA)) + 3))) ';' (Stop SCM+FSA)) ';' (Goto ((card (I ';' (Goto 0))) + 1))) ';' I) ';' ((Goto 0) ';' (Stop SCM+FSA)) by SCMFSA6A:67 ;
then card (if>0 (a,(I ';' (Goto 0)),(Stop SCM+FSA))) = (card ((((Macro (a >0_goto ((card (Stop SCM+FSA)) + 3))) ';' (Stop SCM+FSA)) ';' (Goto ((card (I ';' (Goto 0))) + 1))) ';' I)) + (card ((Goto 0) ';' (Stop SCM+FSA))) by SCMFSA6A:61;
then A3: card ((((Macro (a >0_goto ((card (Stop SCM+FSA)) + 3))) ';' (Stop SCM+FSA)) ';' (Goto ((card (I ';' (Goto 0))) + 1))) ';' I) = (card (if>0 (a,(I ';' (Goto 0)),(Stop SCM+FSA)))) - (card ((Goto 0) ';' (Stop SCM+FSA)))
.= ((card I) + 6) - 2 by A1, SCMFSA_9:2
.= (card I) + 4 ;
then A4: not (card I) + 4 in dom ((((Macro (a >0_goto ((card (Stop SCM+FSA)) + 3))) ';' (Stop SCM+FSA)) ';' (Goto ((card (I ';' (Goto 0))) + 1))) ';' I) ;
set GJ = (Goto 0) ';' (Stop SCM+FSA);
A5: InsCode (goto 0) = 6 by SCMFSA_2:47;
A6: Goto 0 = 0 .--> (goto 0) by SCMFSA8A:def 2;
then A7: (Goto 0) . 0 = goto 0 by FUNCOP_1:87;
dom (Goto 0) = {0} by A6, FUNCOP_1:19;
then A8: 0 in dom (Goto 0) by TARSKI:def 1;
A9: dom (Goto 0) c= dom ((Goto 0) ';' (Stop SCM+FSA)) by SCMFSA6A:56;
then 0 + ((card I) + 4) in { (il + ((card I) + 4)) where il is Element of NAT : il in dom ((Goto 0) ';' (Stop SCM+FSA)) } by A8;
then A10: (card I) + 4 in dom (Shift (((Goto 0) ';' (Stop SCM+FSA)),((card I) + 4))) by VALUED_1:def 12;
then A11: (Shift (((Goto 0) ';' (Stop SCM+FSA)),((card I) + 4))) /. ((card I) + 4) = (Shift (((Goto 0) ';' (Stop SCM+FSA)),((card I) + 4))) . (0 + ((card I) + 4)) by PARTFUN1:def 8
.= ((Goto 0) ';' (Stop SCM+FSA)) . 0 by A8, A9, VALUED_1:def 12
.= goto 0 by A8, A7, A5, SCMFSA6A:54 ;
x: card (Stop SCM+FSA) = 1 by COMPOS_1:46;
card (I ';' (Goto 0)) = (card I) + (card (Goto 0)) by SCMFSA6A:61
.= (card I) + 1 by SCMFSA8A:29 ;
then ((card (I ';' (Goto 0))) + (card (Stop SCM+FSA))) + 3 = ((card I) + 4) + 1 by x;
then (card I) + 4 < ((card (I ';' (Goto 0))) + (card (Stop SCM+FSA))) + 3 by NAT_1:13;
hence A12: (card I) + 4 in dom (if>0 (a,(I ';' (Goto 0)),(Stop SCM+FSA))) by SCMFSA8C:57; :: thesis: (if>0 (a,(I ';' (Goto 0)),(Stop SCM+FSA))) . ((card I) + 4) = goto (0 + ((card I) + 4))
A13: dom (if>0 (a,(I ';' (Goto 0)),(Stop SCM+FSA))) = (dom (Directed ((((Macro (a >0_goto ((card (Stop SCM+FSA)) + 3))) ';' (Stop SCM+FSA)) ';' (Goto ((card (I ';' (Goto 0))) + 1))) ';' I))) \/ (dom (ProgramPart (Relocated (((Goto 0) ';' (Stop SCM+FSA)),((card I) + 4))))) by A2, A3, FUNCT_4:def 1;
then dom (if>0 (a,(I ';' (Goto 0)),(Stop SCM+FSA))) = (dom ((((Macro (a >0_goto ((card (Stop SCM+FSA)) + 3))) ';' (Stop SCM+FSA)) ';' (Goto ((card (I ';' (Goto 0))) + 1))) ';' I)) \/ (dom (ProgramPart (Relocated (((Goto 0) ';' (Stop SCM+FSA)),((card I) + 4))))) by FUNCT_4:105;
then (card I) + 4 in dom (ProgramPart (Relocated (((Goto 0) ';' (Stop SCM+FSA)),((card I) + 4)))) by A12, A4, XBOOLE_0:def 3;
hence (if>0 (a,(I ';' (Goto 0)),(Stop SCM+FSA))) . ((card I) + 4) = (ProgramPart (Relocated (((Goto 0) ';' (Stop SCM+FSA)),((card I) + 4)))) . ((card I) + 4) by A12, A2, A3, A13, FUNCT_4:def 1
.= (Reloc ((ProgramPart ((Goto 0) ';' (Stop SCM+FSA))),((card I) + 4))) . ((card I) + 4) by COMPOS_1:116
.= (Reloc (((Goto 0) ';' (Stop SCM+FSA)),((card I) + 4))) . ((card I) + 4) by RELAT_1:209
.= IncAddr ((goto 0),((card I) + 4)) by A10, A11, SCMFSA_4:24
.= goto (0 + ((card I) + 4)) by SCMFSA_4:14 ;
:: thesis: verum