let N be with_zero set ; :: thesis: 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; :: thesis: 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; :: thesis: for k being Nat holds Start-At (((IC p) -' k),S) c= DecIC (p,k)
let k be Nat; :: thesis: 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; :: thesis: verum