let i1, il be Instruction-Location of SCM+FSA ; :: thesis: for a being Int-Location holds NIC (a >0_goto i1),il = {i1,(Next il)}
let a be Int-Location ; :: thesis: NIC (a >0_goto i1),il = {i1,(Next il)}

hereby :: according to TARSKI:def 3,XBOOLE_0:def 10 :: thesis: {i1,(Next 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,(Next il)} )
assume x in NIC (a >0_goto i1),il ; :: thesis: b₁ in {i1,(Next il)}
then consider s being State of SCM+FSA such that
A1: ( x = IC (Following s) & IC s = il & s . il = a >0_goto i1 ) ;

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

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

then x = i1 by A1, SCMFSA_2:97;
hence x in {i1,(Next il)} by TARSKI:def 2; :: thesis: verum

end;

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

then x = Next il by A1, SCMFSA_2:97;
hence x in {i1,(Next il)} by TARSKI:def 2; :: thesis: verum

end;

end;

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in {i1,(Next il)} or x in NIC (a >0_goto i1),il )
assume A2: x in {i1,(Next il)} ; :: thesis: x in NIC (a >0_goto i1),il
consider t being State of SCM+FSA ;
il in NAT by AMI_1:def 4;
then reconsider il1 = il as Element of ObjectKind (IC SCM+FSA ) by AMI_1:def 11;
reconsider I = a >0_goto i1 as Element of ObjectKind il by AMI_1:def 14;
set u = t +* ((IC SCM+FSA ),il --> il1,I);
il in NAT by AMI_1:def 4;
then A3: a <> il by Th3;
A4: IC SCM+FSA <> a by SCMFSA_2:81;

per cases ( x = i1 or x = Next il ) by A2, TARSKI:def 2;

suppose A5: x = i1 ; :: thesis: x in NIC (a >0_goto i1),il

set v = (t +* ((IC SCM+FSA ),il --> il1,I)) +* (a .--> 1);
A6: dom (a .--> 1) = {a} by FUNCOP_1:19;
then not IC SCM+FSA in dom (a .--> 1) by A4, TARSKI:def 1;
then A7: IC ((t +* ((IC SCM+FSA ),il --> il1,I)) +* (a .--> 1)) = IC (t +* ((IC SCM+FSA ),il --> il1,I)) by FUNCT_4:12
.= il by AMI_1:129 ;
not il in dom (a .--> 1) by A3, A6, TARSKI:def 1;
then A8: ((t +* ((IC SCM+FSA ),il --> il1,I)) +* (a .--> 1)) . il = (t +* ((IC SCM+FSA ),il --> il1,I)) . il by FUNCT_4:12
.= I by AMI_1:129 ;
a in dom (a .--> 1) by A6, TARSKI:def 1;
then ((t +* ((IC SCM+FSA ),il --> il1,I)) +* (a .--> 1)) . a = (a .--> 1) . a by FUNCT_4:14
.= 1 by FUNCOP_1:87 ;
then IC (Following ((t +* ((IC SCM+FSA ),il --> il1,I)) +* (a .--> 1))) = i1 by A7, A8, SCMFSA_2:97;
hence x in NIC (a >0_goto i1),il by A5, A7, A8; :: thesis: verum

end;

suppose A9: x = Next il ; :: thesis: x in NIC (a >0_goto i1),il

set v = (t +* ((IC SCM+FSA ),il --> il1,I)) +* (a .--> 0 );
A10: dom (a .--> 0 ) = {a} by FUNCOP_1:19;
then not IC SCM+FSA in dom (a .--> 0 ) by A4, TARSKI:def 1;
then A11: IC ((t +* ((IC SCM+FSA ),il --> il1,I)) +* (a .--> 0 )) = IC (t +* ((IC SCM+FSA ),il --> il1,I)) by FUNCT_4:12
.= il by AMI_1:129 ;
not il in dom (a .--> 0 ) by A3, A10, TARSKI:def 1;
then A12: ((t +* ((IC SCM+FSA ),il --> il1,I)) +* (a .--> 0 )) . il = (t +* ((IC SCM+FSA ),il --> il1,I)) . il by FUNCT_4:12
.= I by AMI_1:129 ;
a in dom (a .--> 0 ) by A10, TARSKI:def 1;
then ((t +* ((IC SCM+FSA ),il --> il1,I)) +* (a .--> 0 )) . a = (a .--> 0 ) . a by FUNCT_4:14
.= 0 by FUNCOP_1:87 ;
then IC (Following ((t +* ((IC SCM+FSA ),il --> il1,I)) +* (a .--> 0 ))) = Next il by A11, A12, SCMFSA_2:97;
hence x in NIC (a >0_goto i1),il by A9, A11, A12; :: thesis: verum

end;

end;