let a be Data-Location; :: thesis: for il, k being Nat holds NIC ((a >0_goto k),il) = {k,(il + 1)}
let il, k be Nat; :: thesis: NIC ((a >0_goto k),il) = {k,(il + 1)}
set t = the State of SCM;
set Q = the Instruction-Sequence of SCM;

hereby :: according to TARSKI:def 3,XBOOLE_0:def 10 :: thesis: {k,(il + 1)} c= NIC ((a >0_goto k),il)

let x be object ; :: thesis: ( x in NIC ((a >0_goto k),il) implies b₁ in {k,(il + 1)} )
assume x in NIC ((a >0_goto k),il) ; :: thesis: b₁ in {k,(il + 1)}
then consider s being Element of product (the_Values_of SCM) such that
A1: ( x = IC (Exec ((a >0_goto k),s)) & IC s = il ) ;

per cases ( s . a > 0 or s . a <= 0 ) ;

suppose s . a > 0 ; :: thesis: b₁ in {k,(il + 1)}

then x = k by A1, AMI_3:9;
hence x in {k,(il + 1)} by TARSKI:def 2; :: thesis: verum

end;

suppose s . a <= 0 ; :: thesis: b₁ in {k,(il + 1)}

then x = il + 1 by A1, AMI_3:9;
hence x in {k,(il + 1)} by TARSKI:def 2; :: thesis: verum

end;

end;

end;

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {k,(il + 1)} or x in NIC ((a >0_goto k),il) )
set I = a >0_goto k;
A2: IC <> a by AMI_5:2;
assume A3: x in {k,(il + 1)} ; :: thesis: x in NIC ((a >0_goto k),il)
reconsider n = il as Element of NAT by ORDINAL1:def 12;
reconsider il1 = n as Element of Values (IC ) by MEMSTR_0:def 6;
reconsider u = the State of SCM +* ((IC ),il1) as Element of product (the_Values_of SCM) by CARD_3:107;
reconsider P = the Instruction-Sequence of SCM +* (il,(a >0_goto k)) as Instruction-Sequence of SCM ;

per cases ( x = k or x = il + 1 ) by A3, TARSKI:def 2;

suppose A4: x = k ; :: thesis: x in NIC ((a >0_goto k),il)

reconsider v = u +* (a .--> 1) as Element of product (the_Values_of SCM) by CARD_3:107;
A5: IC in dom the State of SCM by MEMSTR_0:2;
not IC in dom (a .--> 1) by A2, TARSKI:def 1;
then A7: IC v = IC u by FUNCT_4:11
.= n by A5, FUNCT_7:31 ;
reconsider il = il as Element of NAT by ORDINAL1:def 12;
A8: P /. il = P . il by PBOOLE:143;
il in NAT ;
then il in dom the Instruction-Sequence of SCM by PARTFUN1:def 2;
then A9: P . il = a >0_goto k by FUNCT_7:31;
a in dom (a .--> 1) by TARSKI:def 1;
then v . a = (a .--> 1) . a by FUNCT_4:13
.= 1 by FUNCOP_1:72 ;
then IC (Following (P,v)) = k by A7, A9, A8, AMI_3:9;
hence x in NIC ((a >0_goto k),il) by A4, A7, A9, A8; :: thesis: verum

end;

suppose A10: x = il + 1 ; :: thesis: x in NIC ((a >0_goto k),il)

reconsider v = u +* (a .--> 0) as Element of product (the_Values_of SCM) by CARD_3:107;
A11: IC in dom the State of SCM by MEMSTR_0:2;
not IC in dom (a .--> 0) by A2, TARSKI:def 1;
then A13: IC v = IC u by FUNCT_4:11
.= n by A11, FUNCT_7:31 ;
reconsider il = il as Element of NAT by ORDINAL1:def 12;
A14: P /. il = P . il by PBOOLE:143;
il in NAT ;
then il in dom the Instruction-Sequence of SCM by PARTFUN1:def 2;
then A15: P . il = a >0_goto k by FUNCT_7:31;
a in dom (a .--> 0) by TARSKI:def 1;
then v . a = (a .--> 0) . a by FUNCT_4:13
.= 0 by FUNCOP_1:72 ;
then IC (Following (P,v)) = il + 1 by A13, A15, A14, AMI_3:9;
hence x in NIC ((a >0_goto k),il) by A10, A13, A15, A14; :: thesis: verum

end;

end;