let loc be Instruction-Location of SCM ; :: thesis: not goto loc is halting
assume A1: goto loc is halting ; :: thesis: contradiction
reconsider V = goto loc as Element of SCM-Instr ;
reconsider a3 = loc as Element of NAT by AMI_1:def 4;
consider s being SCM-State;
set t = s +* (NAT .--> (succ a3));
set f = the Object-Kind of SCM ;
A2: dom (NAT .--> (succ a3)) = {NAT } by FUNCOP_1:19;
then NAT in dom (NAT .--> (succ a3)) by TARSKI:def 1;
then A3: (s +* (NAT .--> (succ a3))) . NAT = (NAT .--> (succ a3)) . NAT by FUNCT_4:14
.= succ a3 by FUNCOP_1:87 ;
A4: {NAT } c= SCM-Memory by AMI_2:30, ZFMISC_1:37;
A5: dom s = dom SCM-OK by CARD_3:18;
A6: dom (s +* (NAT .--> (succ a3))) = (dom s) \/ (dom (NAT .--> (succ a3))) by FUNCT_4:def 1
.= SCM-Memory \/ (dom (NAT .--> (succ a3))) by A5, FUNCT_2:def 1
.= SCM-Memory \/ {NAT } by FUNCOP_1:19
.= SCM-Memory by A4, XBOOLE_1:12 ;
A7: dom the Object-Kind of SCM = SCM-Memory by FUNCT_2:def 1;
for x being set st x in dom the Object-Kind of SCM holds
(s +* (NAT .--> (succ a3))) . x in the Object-Kind of SCM . x

proof

let x be set ; :: thesis: ( x in dom the Object-Kind of SCM implies (s +* (NAT .--> (succ a3))) . x in the Object-Kind of SCM . x )
assume A8: x in dom the Object-Kind of SCM ; :: thesis: (s +* (NAT .--> (succ a3))) . x in the Object-Kind of SCM . x

per cases ( x = NAT or x <> NAT ) ;

suppose A9: x = NAT ; :: thesis: (s +* (NAT .--> (succ a3))) . x in the Object-Kind of SCM . x

then the Object-Kind of SCM . x = NAT by AMI_2:7, AMI_2:30;
hence (s +* (NAT .--> (succ a3))) . x in the Object-Kind of SCM . x by A3, A9; :: thesis: verum

end;

suppose x <> NAT ; :: thesis: (s +* (NAT .--> (succ a3))) . x in the Object-Kind of SCM . x

then not x in dom (NAT .--> (succ a3)) by A2, TARSKI:def 1;
then (s +* (NAT .--> (succ a3))) . x = s . x by FUNCT_4:12;
hence (s +* (NAT .--> (succ a3))) . x in the Object-Kind of SCM . x by A8, CARD_3:18; :: thesis: verum

end;

end;

end;

then reconsider t = s +* (NAT .--> (succ a3)) as State of SCM by A6, A7, CARD_3:18;
reconsider w = t as SCM-State ;
dom (NAT .--> loc) = {NAT } by FUNCOP_1:19;
then NAT in dom (NAT .--> loc) by TARSKI:def 1;
then A10: (w +* (NAT .--> loc)) . NAT = (NAT .--> loc) . NAT by FUNCT_4:14
.= loc by FUNCOP_1:87 ;
6 is Element of Segm 9 by GR_CY_1:10;
then w +* (NAT .--> loc) = SCM-Chg w,(V jump_address ) by AMI_2:24
.= SCM-Exec-Res V,w by AMI_2:def 16
.= Exec (goto loc),t by AMI_2:def 17
.= t by A1, AMI_1:def 8 ;
hence contradiction by A3, A10; :: thesis: verum