let a be Data-Location ; :: thesis: for il being Element of NAT
for k being natural number holds NIC (a >0_goto k),il = {k,(succ il)}
let il be Element of NAT ; :: thesis: for k being natural number holds NIC (a >0_goto k),il = {k,(succ il)}
let k be natural number ; :: thesis: NIC (a >0_goto k),il = {k,(succ il)}
consider t being State of SCM ;

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

let x be set ; :: thesis: ( x in NIC (a >0_goto k),il implies b₁ in {k,(succ il)} )
assume x in NIC (a >0_goto k),il ; :: thesis: b₁ in {k,(succ il)}
then consider s being Element of product the Object-Kind of SCM such that
A1: ( x = IC (Following (ProgramPart s),s) & IC s = il & (ProgramPart s) /. il = a >0_goto k ) ;

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

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

then x = k by A1, AMI_3:15;
hence x in {k,(succ il)} by TARSKI:def 2; :: thesis: verum

end;

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

then x = succ il by A1, AMI_3:15;
hence x in {k,(succ il)} by TARSKI:def 2; :: thesis: verum

end;

end;

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in {k,(succ il)} or x in NIC (a >0_goto k),il )
reconsider I = a >0_goto k as Element of the Object-Kind of SCM . il by COMPOS_1:def 8;
A2: IC SCM <> a by AMI_5:20;
reconsider il1 = il as Element of ObjectKind (IC SCM ) by COMPOS_1:def 6;
reconsider p = (IC SCM ),il --> il1,I as PartState of SCM by COMPOS_1:37;
reconsider u = t +* p as State of SCM ;
A3: a <> il by Th1;
reconsider n = il1 as Element of NAT ;
assume A4: x in {k,(succ il)} ; :: thesis: x in NIC (a >0_goto k),il

per cases ( x = k or x = succ il ) by A4, TARSKI:def 2;

suppose A5: x = k ; :: thesis: x in NIC (a >0_goto k),il

reconsider v = u +* (a .--> 1) as Element of product the Object-Kind of SCM by PBOOLE:155;
A6: dom (a .--> 1) = {a} by FUNCOP_1:19;
then not IC SCM in dom (a .--> 1) by A2, TARSKI:def 1;
then A7: IC v = IC u by FUNCT_4:12
.= n by AMI_1:129 ;
U: not il in dom (a .--> 1) by A3, A6, TARSKI:def 1;
A8: (ProgramPart v) /. il = v . il by COMPOS_1:38
.= u . n by U, FUNCT_4:12
.= I by AMI_1:129 ;
a in dom (a .--> 1) by A6, TARSKI:def 1;
then v . a = (a .--> 1) . a by FUNCT_4:14
.= 1 by FUNCOP_1:87 ;
then IC (Following (ProgramPart v),v) = k by A7, A8, AMI_3:15;
hence x in NIC (a >0_goto k),il by A5, A7, A8; :: thesis: verum

end;

suppose A9: x = succ il ; :: thesis: x in NIC (a >0_goto k),il

reconsider v = u +* (a .--> 0 ) as Element of product the Object-Kind of SCM by PBOOLE:155;
A10: dom (a .--> 0 ) = {a} by FUNCOP_1:19;
then not IC SCM in dom (a .--> 0 ) by A2, TARSKI:def 1;
then A11: IC v = IC u by FUNCT_4:12
.= n by AMI_1:129 ;
U: not il in dom (a .--> 0 ) by A3, A10, TARSKI:def 1;
A12: (ProgramPart v) /. il = v . il by COMPOS_1:38
.= u . n by U, FUNCT_4:12
.= I by AMI_1:129 ;
a in dom (a .--> 0 ) by A10, TARSKI:def 1;
then v . a = (a .--> 0 ) . a by FUNCT_4:14
.= 0 by FUNCOP_1:87 ;
then IC (Following (ProgramPart v),v) = succ il by A11, A12, AMI_3:15;
hence x in NIC (a >0_goto k),il by A9, A11, A12; :: thesis: verum

end;

end;