let il be Element of NAT ; :: thesis: SUCC (il,SCM) = {il,(succ il)}
set X = { ((NIC (I,il)) \ (JUMP I)) where I is Element of the Instructions of SCM : verum } ;
set N = {il,(succ il)};

now

let x be set ; :: thesis: ( ( x in union { ((NIC (I,il)) \ (JUMP I)) where I is Element of the Instructions of SCM : verum } implies x in {il,(succ il)} ) & ( x in {il,(succ il)} implies b₁ in union { ((NIC (b₂,il)) \ (JUMP b₂)) where I is Element of the Instructions of SCM : verum } ) )

hereby :: thesis: ( x in {il,(succ il)} implies b₁ in union { ((NIC (b₂,il)) \ (JUMP b₂)) where I is Element of the Instructions of SCM : verum } )

assume x in union { ((NIC (I,il)) \ (JUMP I)) where I is Element of the Instructions of SCM : verum } ; :: thesis: x in {il,(succ il)}
then consider Y being set such that
A1: x in Y and
A2: Y in { ((NIC (I,il)) \ (JUMP I)) where I is Element of the Instructions of SCM : verum } by TARSKI:def 4;
consider i being Element of the Instructions of SCM such that
A3: Y = (NIC (i,il)) \ (JUMP i) by A2;

per cases ( i = [0,{},{}] or ex a, b being Data-Location st i = a := b or ex a, b being Data-Location st i = AddTo (a,b) or ex a, b being Data-Location st i = SubFrom (a,b) or ex a, b being Data-Location st i = MultBy (a,b) or ex a, b being Data-Location st i = Divide (a,b) or ex k being natural number st i = SCM-goto k or ex a being Data-Location ex k being natural number st i = a =0_goto k or ex a being Data-Location ex k being natural number st i = a >0_goto k ) by AMI_3:69;

suppose i = [0,{},{}] ; :: thesis: x in {il,(succ il)}

then x in {il} \ (JUMP (halt SCM)) by A1, A3, Th40, AMI_3:71;
then x = il by TARSKI:def 1;
hence x in {il,(succ il)} by TARSKI:def 2; :: thesis: verum

end;

suppose ex a, b being Data-Location st i = a := b ; :: thesis: x in {il,(succ il)}

then consider a, b being Data-Location such that
A4: i = a := b ;
x in {(succ il)} \ (JUMP (a := b)) by A1, A3, A4, Th41;
then x = succ il by TARSKI:def 1;
hence x in {il,(succ il)} by TARSKI:def 2; :: thesis: verum

end;

suppose ex a, b being Data-Location st i = AddTo (a,b) ; :: thesis: x in {il,(succ il)}

then consider a, b being Data-Location such that
A5: i = AddTo (a,b) ;
x in {(succ il)} \ (JUMP (AddTo (a,b))) by A1, A3, A5, Th42;
then x = succ il by TARSKI:def 1;
hence x in {il,(succ il)} by TARSKI:def 2; :: thesis: verum

end;

suppose ex a, b being Data-Location st i = SubFrom (a,b) ; :: thesis: x in {il,(succ il)}

then consider a, b being Data-Location such that
A6: i = SubFrom (a,b) ;
x in {(succ il)} \ (JUMP (SubFrom (a,b))) by A1, A3, A6, Th43;
then x = succ il by TARSKI:def 1;
hence x in {il,(succ il)} by TARSKI:def 2; :: thesis: verum

end;

suppose ex a, b being Data-Location st i = MultBy (a,b) ; :: thesis: x in {il,(succ il)}

then consider a, b being Data-Location such that
A7: i = MultBy (a,b) ;
x in {(succ il)} \ (JUMP (MultBy (a,b))) by A1, A3, A7, Th44;
then x = succ il by TARSKI:def 1;
hence x in {il,(succ il)} by TARSKI:def 2; :: thesis: verum

end;

suppose ex a, b being Data-Location st i = Divide (a,b) ; :: thesis: x in {il,(succ il)}

then consider a, b being Data-Location such that
A8: i = Divide (a,b) ;
x in {(succ il)} \ (JUMP (Divide (a,b))) by A1, A3, A8, Th45;
then x = succ il by TARSKI:def 1;
hence x in {il,(succ il)} by TARSKI:def 2; :: thesis: verum

end;

suppose ex k being natural number st i = SCM-goto k ; :: thesis: x in {il,(succ il)}

then consider k being natural number such that
A9: i = SCM-goto k ;
x in {k} \ (JUMP i) by A1, A3, A9, Th46;
then x in {k} \ {k} by A9, Th47;
hence x in {il,(succ il)} by XBOOLE_1:37; :: thesis: verum

end;

suppose ex a being Data-Location ex k being natural number st i = a =0_goto k ; :: thesis: x in {il,(succ il)}

then consider a being Data-Location , k being natural number such that
A10: i = a =0_goto k ;
A11: NIC (i,il) = {k,(succ il)} by A10, Th48;
x in NIC (i,il) by A1, A3, XBOOLE_0:def 5;
then A12: ( x = k or x = succ il ) by A11, TARSKI:def 2;
x in (NIC (i,il)) \ {k} by A1, A3, A10, Th49;
then not x in {k} by XBOOLE_0:def 5;
hence x in {il,(succ il)} by A12, TARSKI:def 1, TARSKI:def 2; :: thesis: verum

end;

suppose ex a being Data-Location ex k being natural number st i = a >0_goto k ; :: thesis: x in {il,(succ il)}

then consider a being Data-Location , k being natural number such that
A13: i = a >0_goto k ;
A14: NIC (i,il) = {k,(succ il)} by A13, Th50;
x in NIC (i,il) by A1, A3, XBOOLE_0:def 5;
then A15: ( x = k or x = succ il ) by A14, TARSKI:def 2;
x in (NIC (i,il)) \ {k} by A1, A3, A13, Th51;
then not x in {k} by XBOOLE_0:def 5;
hence x in {il,(succ il)} by A15, TARSKI:def 1, TARSKI:def 2; :: thesis: verum

end;

end;

end;

assume A16: x in {il,(succ il)} ; :: thesis: b₁ in union { ((NIC (b₂,il)) \ (JUMP b₂)) where I is Element of the Instructions of SCM : verum }

per cases ( x = il or x = succ il ) by A16, TARSKI:def 2;

suppose A17: x = il ; :: thesis: b₁ in union { ((NIC (b₂,il)) \ (JUMP b₂)) where I is Element of the Instructions of SCM : verum }

set i = halt SCM;
(NIC ((halt SCM),il)) \ (JUMP (halt SCM)) = {il} by Th40;
then A18: {il} in { ((NIC (I,il)) \ (JUMP I)) where I is Element of the Instructions of SCM : verum } ;
x in {il} by A17, TARSKI:def 1;
hence x in union { ((NIC (I,il)) \ (JUMP I)) where I is Element of the Instructions of SCM : verum } by A18, TARSKI:def 4; :: thesis: verum

end;

suppose A19: x = succ il ; :: thesis: b₁ in union { ((NIC (b₂,il)) \ (JUMP b₂)) where I is Element of the Instructions of SCM : verum }

consider a, b being Data-Location ;
set i = AddTo (a,b);
(NIC ((AddTo (a,b)),il)) \ (JUMP (AddTo (a,b))) = {(succ il)} by Th42;
then A20: {(succ il)} in { ((NIC (I,il)) \ (JUMP I)) where I is Element of the Instructions of SCM : verum } ;
x in {(succ il)} by A19, TARSKI:def 1;
hence x in union { ((NIC (I,il)) \ (JUMP I)) where I is Element of the Instructions of SCM : verum } by A20, TARSKI:def 4; :: thesis: verum

end;

end;

end;

hence SUCC (il,SCM) = {il,(succ il)} by TARSKI:2; :: thesis: verum