let n be Element of NAT ; :: thesis: for I being Program of SCMPDS holds (I +* (Start-At n,SCMPDS )) | NAT = I
let I be Program of SCMPDS ; :: thesis: (I +* (Start-At n,SCMPDS )) | NAT = I
NAT misses dom (Start-At n,SCMPDS )

proof

assume not NAT misses dom (Start-At n,SCMPDS ) ; :: thesis: contradiction
then consider x being set such that
A1: x in NAT and
A2: x in dom (Start-At n,SCMPDS ) by XBOOLE_0:3;
A3: x is Element of NAT by A1;
dom (Start-At n,SCMPDS ) = {(IC SCMPDS )} by FUNCOP_1:19;
then x = IC SCMPDS by A2, TARSKI:def 1;
hence contradiction by A3, COMPOS_1:3; :: thesis: verum

end;

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