let p be autonomic FinPartState of SCMPDS ; :: thesis: ( IC SCMPDS in dom p implies IC p in dom p )
assume A1: IC SCMPDS in dom p ; :: thesis: IC p in dom p
assume A2: not IC p in dom p ; :: thesis: contradiction
set il = IC p;
set p1 = p +* ((IC p) .--> (goto 0 ));
set p2 = p +* ((IC p) .--> (goto 1));
consider s1 being State of SCMPDS such that
A3: p +* ((IC p) .--> (goto 0 )) c= s1 by CARD_3:97;
consider s2 being State of SCMPDS such that
A4: p +* ((IC p) .--> (goto 1)) c= s2 by CARD_3:97;
not p is autonomic

proof

A5: dom ((IC p) .--> (goto 1)) = {(IC p)} by FUNCOP_1:19;
A6: dom ((IC p) .--> (goto 0 )) = {(IC p)} by FUNCOP_1:19;
take s1 ; :: according to AMI_1:def 25 :: thesis: ex b₁ being Element of product the Object-Kind of SCMPDS st
( p c= s1 & p c= b₁ & not for b₂ being Element of NAT holds (Computation s1,b₂) | (dom p) = (Computation b₁,b₂) | (dom p) )
take s2 ; :: thesis: ( p c= s1 & p c= s2 & not for b₁ being Element of NAT holds (Computation s1,b₁) | (dom p) = (Computation s2,b₁) | (dom p) )
dom p misses {(IC p)} by A2, ZFMISC_1:56;
then A7: ( p c= p +* ((IC p) .--> (goto 0 )) & p c= p +* ((IC p) .--> (goto 1)) ) by A5, A6, FUNCT_4:33;
hence ( p c= s1 & p c= s2 ) by A3, A4, XBOOLE_1:1; :: thesis: not for b₁ being Element of NAT holds (Computation s1,b₁) | (dom p) = (Computation s2,b₁) | (dom p)
take 1 ; :: thesis: not (Computation s1,1) | (dom p) = (Computation s2,1) | (dom p)
A8: IC p in dom ((IC p) .--> (goto 1)) by A5, TARSKI:def 1;
A9: IC p in dom ((IC p) .--> (goto 0 )) by A6, TARSKI:def 1;
dom (p +* ((IC p) .--> (goto 0 ))) = (dom p) \/ (dom ((IC p) .--> (goto 0 ))) by FUNCT_4:def 1;
then IC p in dom (p +* ((IC p) .--> (goto 0 ))) by A9, XBOOLE_0:def 3;
then A10: s1 . (IC p) = (p +* ((IC p) .--> (goto 0 ))) . (IC p) by A3, GRFUNC_1:8
.= ((IC p) .--> (goto 0 )) . (IC p) by A9, FUNCT_4:14
.= goto 0 by FUNCOP_1:87 ;
dom (p +* ((IC p) .--> (goto 1))) = (dom p) \/ (dom ((IC p) .--> (goto 1))) by FUNCT_4:def 1;
then IC p in dom (p +* ((IC p) .--> (goto 1))) by A8, XBOOLE_0:def 3;
then A11: s2 . (IC p) = (p +* ((IC p) .--> (goto 1))) . (IC p) by A4, GRFUNC_1:8
.= ((IC p) .--> (goto 1)) . (IC p) by A8, FUNCT_4:14
.= goto 1 by FUNCOP_1:87 ;
A12: (Following s1) . (IC SCMPDS ) = (Exec (goto 0 ),s1) . (IC SCMPDS ) by A1, A3, A7, A10, AMI_1:97, XBOOLE_1:1
.= ICplusConst s1,0 by SCMPDS_2:66 ;
A13: (Following s2) . (IC SCMPDS ) = (Exec (goto 1),s2) . (IC SCMPDS ) by A1, A4, A7, A11, AMI_1:97, XBOOLE_1:1
.= ICplusConst s2,1 by SCMPDS_2:66 ;
assume A14: (Computation s1,1) | (dom p) = (Computation s2,1) | (dom p) ; :: thesis: contradiction
A15: (Following s1) | (dom p) = (Following (Computation s1,0 )) | (dom p) by AMI_1:13
.= (Computation s1,(0 + 1)) | (dom p) by AMI_1:14
.= (Following (Computation s2,0 )) | (dom p) by A14, AMI_1:14
.= (Following s2) | (dom p) by AMI_1:13 ;
A16: ICplusConst s1,0 = ((Following s1) | (dom p)) . (IC SCMPDS ) by A1, A12, FUNCT_1:72
.= ICplusConst s2,1 by A1, A13, A15, FUNCT_1:72 ;
IC s2 = IC p by A1, A4, A7, AMI_1:97, XBOOLE_1:1
.= IC s1 by A1, A3, A7, AMI_1:97, XBOOLE_1:1 ;
hence contradiction by A16, Th19; :: thesis: verum

end;

hence contradiction ; :: thesis: verum