let N be non empty with_non-empty_elements set ; :: thesis: for n being Element of NAT
for S being non empty stored-program IC-Ins-separated definite realistic COM-Struct of N
for I being Program of S holds (I +* (Start-At (n,S))) | NAT = I
let n be Element of NAT ; :: thesis: for S being non empty stored-program IC-Ins-separated definite realistic COM-Struct of N
for I being Program of S holds (I +* (Start-At (n,S))) | NAT = I
let S be non empty stored-program IC-Ins-separated definite realistic COM-Struct of N; :: thesis: for I being Program of S holds (I +* (Start-At (n,S))) | NAT = I
let I be Program of S; :: thesis: (I +* (Start-At (n,S))) | NAT = I
NAT misses dom (Start-At (n,S))

proof

assume not NAT misses dom (Start-At (n,S)) ; :: thesis: contradiction
then consider x being set such that
A1: x in NAT and
A2: x in dom (Start-At (n,S)) by XBOOLE_0:3;
dom (Start-At (n,S)) = {(IC S)} by FUNCOP_1:19;
then x = IC S by A2, TARSKI:def 1;
hence contradiction by A1, Th3; :: thesis: verum

end;

then ( dom I c= NAT & (I +* (Start-At (n,S))) | NAT = I | NAT ) by FUNCT_4:76;
hence (I +* (Start-At (n,S))) | NAT = I by RELAT_1:97; :: thesis: verum