let N be with_zero set ; for S being non empty with_non-empty_values IC-Ins-separated Mem-Struct over N
for p being PartState of S
for k being Nat holds Start-At (((IC p) -' k),S) c= DecIC (p,k)
let S be non empty with_non-empty_values IC-Ins-separated Mem-Struct over N; for p being PartState of S
for k being Nat holds Start-At (((IC p) -' k),S) c= DecIC (p,k)
let p be PartState of S; for k being Nat holds Start-At (((IC p) -' k),S) c= DecIC (p,k)
let k be Nat; Start-At (((IC p) -' k),S) c= DecIC (p,k)
A1:
IC (DecIC (p,k)) = (IC p) -' k
by Th65;
A2:
IC in dom (DecIC (p,k))
by Th64;
A3:
( Start-At (((IC p) -' k),S) = {[(IC ),((IC p) -' k)]} & [(IC ),((IC p) -' k)] in DecIC (p,k) )
by A2, A1, FUNCT_1:def 2, FUNCT_4:82;
for x being object st x in Start-At (((IC p) -' k),S) holds
x in DecIC (p,k)
by A3, TARSKI:def 1;
hence
Start-At (((IC p) -' k),S) c= DecIC (p,k)
by TARSKI:def 3; verum