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 IC in dom (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 IC in dom (DecIC (p,k))
let p be PartState of S; :: thesis: for k being Nat holds IC in dom (DecIC (p,k))
let k be Nat; :: thesis: IC in dom (DecIC (p,k))
A1: dom (DecIC (p,k)) = (dom p) \/ (dom (Start-At (((IC p) -' k),S))) by FUNCT_4:def 1;
IC in dom (Start-At (((IC p) -' k),S)) by Th15;
hence IC in dom (DecIC (p,k)) by A1, XBOOLE_0:def 3; :: thesis: verum