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)}
set t = the 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 (Exec ((a >0_goto i1),s)) and
A2: IC s = il ;

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

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

then x = i1 by A1, SCMFSA_2:97;
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, SCMFSA_2:97;
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 COMPOS_1:def 8;
A3: IC <> a by SCMFSA_2:81;
reconsider il1 = il as Element of ObjectKind (IC ) by COMPOS_1:def 6;
reconsider n = il as Element of NAT ;
reconsider p = ((IC ),il) --> (il1,I) as PartState of SCM+FSA by COMPOS_1:37;
reconsider u = the State of SCM+FSA +* p as State of SCM+FSA ;
A4: a <> il by Th3;
assume A5: x in {i1,(succ il)} ; :: thesis: x in NIC ((a >0_goto i1),il)

per cases ( x = i1 or x = succ il ) by A5, TARSKI:def 2;

suppose A6: x = i1 ; :: 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;
A7: dom (a .--> 1) = {a} by FUNCOP_1:19;
then not IC in dom (a .--> 1) by A3, TARSKI:def 1;
then A8: IC v = IC u by FUNCT_4:12
.= n by EXTPRO_1:26 ;
A9: not il in dom (a .--> 1) by A4, A7, TARSKI:def 1;
A10: (ProgramPart v) /. il = v . il by COMPOS_1:38
.= u . n by A9, FUNCT_4:12
.= I by EXTPRO_1:26 ;
a in dom (a .--> 1) by A7, TARSKI:def 1;
then v . a = (a .--> 1) . a by FUNCT_4:14
.= 1 by FUNCOP_1:87 ;
then IC (Following ((ProgramPart v),v)) = i1 by A8, A10, SCMFSA_2:97;
hence x in NIC ((a >0_goto i1),il) by A6, A8, A10; :: thesis: verum

end;

suppose A11: x = succ il ; :: 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;
A12: dom (a .--> 0) = {a} by FUNCOP_1:19;
then not IC in dom (a .--> 0) by A3, TARSKI:def 1;
then A13: IC v = IC u by FUNCT_4:12
.= n by EXTPRO_1:26 ;
A14: not il in dom (a .--> 0) by A4, A12, TARSKI:def 1;
A15: (ProgramPart v) /. il = v . il by COMPOS_1:38
.= u . n by A14, FUNCT_4:12
.= I by EXTPRO_1:26 ;
a in dom (a .--> 0) by A12, 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 A13, A15, SCMFSA_2:97;
hence x in NIC ((a >0_goto i1),il) by A11, A13, A15; :: thesis: verum

end;

end;