let a be Data-Location; :: thesis: for k being Nat holds JUMP (a =0_goto k) = {k}
let k be Nat; :: thesis: JUMP (a =0_goto k) = {k}
set X = { (NIC ((a =0_goto k),il)) where il is Nat : verum } ;

now :: thesis: for x being object holds
( ( x in meet { (NIC ((a =0_goto k),il)) where il is Nat : verum } implies x in {k} ) & ( x in {k} implies x in meet { (NIC ((a =0_goto k),il)) where il is Nat : verum } ) )

let x be object ; :: thesis: ( ( x in meet { (NIC ((a =0_goto k),il)) where il is Nat : verum } implies x in {k} ) & ( x in {k} implies x in meet { (NIC ((a =0_goto k),il)) where il is Nat : verum } ) )

A1: now :: thesis: for Y being set st Y in { (NIC ((a =0_goto k),il)) where il is Nat : verum } holds
k in Y

let Y be set ; :: thesis: ( Y in { (NIC ((a =0_goto k),il)) where il is Nat : verum } implies k in Y )
assume Y in { (NIC ((a =0_goto k),il)) where il is Nat : verum } ; :: thesis: k in Y
then consider il being Nat such that
A2: Y = NIC ((a =0_goto k),il) ;
NIC ((a =0_goto k),il) = {k,(il + 1)} by Th17;
hence k in Y by A2, TARSKI:def 2; :: thesis: verum

end;

hereby :: thesis: ( x in {k} implies x in meet { (NIC ((a =0_goto k),il)) where il is Nat : verum } )

set il1 = 1;
set il2 = 2;
assume A3: x in meet { (NIC ((a =0_goto k),il)) where il is Nat : verum } ; :: thesis: x in {k}
A4: NIC ((a =0_goto k),2) = {k,(2 + 1)} by Th17;
NIC ((a =0_goto k),2) in { (NIC ((a =0_goto k),il)) where il is Nat : verum } ;
then x in NIC ((a =0_goto k),2) by A3, SETFAM_1:def 1;
then A5: ( x = k or x = 2 + 1 ) by A4, TARSKI:def 2;
A6: NIC ((a =0_goto k),1) = {k,(1 + 1)} by Th17;
NIC ((a =0_goto k),1) in { (NIC ((a =0_goto k),il)) where il is Nat : verum } ;
then x in NIC ((a =0_goto k),1) by A3, SETFAM_1:def 1;
then ( x = k or x = 1 + 1 ) by A6, TARSKI:def 2;
hence x in {k} by A5, TARSKI:def 1; :: thesis: verum

end;

assume x in {k} ; :: thesis: x in meet { (NIC ((a =0_goto k),il)) where il is Nat : verum }
then A7: x = k by TARSKI:def 1;
reconsider k = k as Element of NAT by ORDINAL1:def 12;
NIC ((a =0_goto k),k) in { (NIC ((a =0_goto k),il)) where il is Nat : verum } ;
hence x in meet { (NIC ((a =0_goto k),il)) where il is Nat : verum } by A7, A1, SETFAM_1:def 1; :: thesis: verum

end;

hence JUMP (a =0_goto k) = {k} by TARSKI:2; :: thesis: verum