let i1, il be Element of NAT ; :: thesis: for a being Int-Location holds NIC (a =0_goto i1),il = {i1,(succ il)}
let a be Int-Location ; :: thesis: NIC (a =0_goto i1),il = {i1,(succ il)}
consider t being State of SCM+FSA ;

hereby :: according to TARSKI:def 3,XBOOLE_0:def 10 :: thesis: {i1,(succ il)} c= NIC (a =0_goto i1),il

let x be set ; :: thesis: ( x in NIC (a =0_goto i1),il implies b₁ in {i1,(succ il)} )
assume x in NIC (a =0_goto i1),il ; :: thesis: b₁ in {i1,(succ il)}
then consider s being Element of product the Object-Kind of SCM+FSA such that
A1: x = IC (Following (ProgramPart s),s) and
A2: IC s = il and
A3: (ProgramPart s) /. il = a =0_goto i1 ;

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

suppose s . a = 0 ; :: thesis: b₁ in {i1,(succ il)}

then x = i1 by A1, A2, A3, SCMFSA_2:96;
hence x in {i1,(succ il)} by TARSKI:def 2; :: thesis: verum

end;

suppose s . a <> 0 ; :: thesis: b₁ in {i1,(succ il)}

then x = succ il by A1, A2, A3, SCMFSA_2:96;
hence x in {i1,(succ il)} by TARSKI:def 2; :: thesis: verum

end;

end;

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in {i1,(succ il)} or x in NIC (a =0_goto i1),il )
reconsider I = a =0_goto i1 as Element of the Object-Kind of SCM+FSA . il by AMI_1:def 14;
A4: IC SCM+FSA <> a by SCMFSA_2:81;
reconsider il1 = il as Element of ObjectKind (IC SCM+FSA ) by AMI_1:def 11;
reconsider p = (IC SCM+FSA ),il --> il1,I as PartState of SCM+FSA by AMI_1:149;
reconsider u = t +* p as State of SCM+FSA ;
A5: a <> il by Th3;
reconsider n = il as Element of NAT ;
assume A6: x in {i1,(succ il)} ; :: thesis: x in NIC (a =0_goto i1),il

per cases ( x = i1 or x = succ il ) by A6, TARSKI:def 2;

suppose A7: x = i1 ; :: thesis: x in NIC (a =0_goto i1),il

reconsider v = u +* (a .--> 0 ) as Element of product the Object-Kind of SCM+FSA by PBOOLE:155;
A8: dom (a .--> 0 ) = {a} by FUNCOP_1:19;
then not IC SCM+FSA in dom (a .--> 0 ) by A4, TARSKI:def 1;
then A9: IC v = IC u by FUNCT_4:12
.= n by AMI_1:129 ;
U: not il in dom (a .--> 0 ) by A5, A8, TARSKI:def 1;
A10: (ProgramPart v) /. il = v . il by AMI_1:150
.= u . n by FUNCT_4:12, U
.= I by AMI_1:129 ;
a in dom (a .--> 0 ) by A8, TARSKI:def 1;
then v . a = (a .--> 0 ) . a by FUNCT_4:14
.= 0 by FUNCOP_1:87 ;
then IC (Following (ProgramPart v),v) = i1 by A9, A10, SCMFSA_2:96;
hence x in NIC (a =0_goto i1),il by A7, A9, A10; :: thesis: verum

end;

suppose A11: x = succ il ; :: thesis: x in NIC (a =0_goto i1),il

reconsider v = u +* (a .--> 1) as Element of product the Object-Kind of SCM+FSA by PBOOLE:155;
A12: dom (a .--> 1) = {a} by FUNCOP_1:19;
then not IC SCM+FSA in dom (a .--> 1) by A4, TARSKI:def 1;
then A13: IC v = IC u by FUNCT_4:12
.= n by AMI_1:129 ;
U: not il in dom (a .--> 1) by A5, A12, TARSKI:def 1;
A14: (ProgramPart v) /. il = v . il by AMI_1:150
.= u . n by FUNCT_4:12, U
.= I by AMI_1:129 ;
a in dom (a .--> 1) by A12, TARSKI:def 1;
then v . a = (a .--> 1) . a by FUNCT_4:14
.= 1 by FUNCOP_1:87 ;
then IC (Following (ProgramPart v),v) = succ il by A13, A14, SCMFSA_2:96;
hence x in NIC (a =0_goto i1),il by A11, A13, A14; :: thesis: verum

end;

end;