let N be non empty with_non-empty_elements set ; :: thesis: for S being non empty stored-program IC-Ins-separated definite realistic COM-Struct of N
for k being natural number
for l1, l2 being Element of NAT holds
( Start-At ((l1 + k),S) = Start-At ((l2 + k),S) iff Start-At (l1,S) = Start-At (l2,S) )
let S be non empty stored-program IC-Ins-separated definite realistic COM-Struct of N; :: thesis: for k being natural number
for l1, l2 being Element of NAT holds
( Start-At ((l1 + k),S) = Start-At ((l2 + k),S) iff Start-At (l1,S) = Start-At (l2,S) )
let k be natural number ; :: thesis: for l1, l2 being Element of NAT holds
( Start-At ((l1 + k),S) = Start-At ((l2 + k),S) iff Start-At (l1,S) = Start-At (l2,S) )
let l1, l2 be Element of NAT ; :: thesis: ( Start-At ((l1 + k),S) = Start-At ((l2 + k),S) iff Start-At (l1,S) = Start-At (l2,S) )

hereby :: thesis: ( Start-At (l1,S) = Start-At (l2,S) implies Start-At ((l1 + k),S) = Start-At ((l2 + k),S) )

assume Start-At ((l1 + k),S) = Start-At ((l2 + k),S) ; :: thesis: Start-At (l1,S) = Start-At (l2,S)
then {[(IC S),(l1 + k)]} = (IC S) .--> (l2 + k) by FUNCT_4:87;
then {[(IC S),(l1 + k)]} = {[(IC S),(l2 + k)]} by FUNCT_4:87;
then [(IC S),(l1 + k)] = [(IC S),(l2 + k)] by ZFMISC_1:6;
then l1 + k = l2 + k by ZFMISC_1:33;
hence Start-At (l1,S) = Start-At (l2,S) ; :: thesis: verum

end;

assume Start-At (l1,S) = Start-At (l2,S) ; :: thesis: Start-At ((l1 + k),S) = Start-At ((l2 + k),S)
then {[(IC S),l1]} = Start-At (l2,S) by FUNCT_4:87;
then {[(IC S),l1]} = {[(IC S),l2]} by FUNCT_4:87;
then [(IC S),l1] = [(IC S),l2] by ZFMISC_1:6;
hence Start-At ((l1 + k),S) = Start-At ((l2 + k),S) by ZFMISC_1:33; :: thesis: verum