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

proof

assume not NAT misses dom (Start-At n,SCM+FSA ) ; :: thesis: contradiction
then A1: ex x being set st
( x in NAT & x in dom (Start-At n,SCM+FSA ) ) by XBOOLE_0:3;
dom (Start-At n,SCM+FSA ) = {(IC SCM+FSA )} by FUNCOP_1:19;
hence contradiction by A1, Lm1, TARSKI:def 1; :: thesis: verum

end;

then ( dom I c= NAT & (I +* (Start-At n,SCM+FSA )) | NAT = I | NAT ) by FUNCT_4:76, RELAT_1:def 18;
hence (I +* (Start-At n,SCM+FSA )) | NAT = I by RELAT_1:97; :: thesis: verum