let N be with_zero set ; :: thesis: for S being non empty with_non-empty_values IC-Ins-separated Mem-Struct over N
for s being State of S
for k being Nat st k <= IC s holds
(IC (DecIC (s,k))) + k = IC s
let S be non empty with_non-empty_values IC-Ins-separated Mem-Struct over N; :: thesis: for s being State of S
for k being Nat st k <= IC s holds
(IC (DecIC (s,k))) + k = IC s
let s be State of S; :: thesis: for k being Nat st k <= IC s holds
(IC (DecIC (s,k))) + k = IC s
let k be Nat; :: thesis: ( k <= IC s implies (IC (DecIC (s,k))) + k = IC s )
assume A1: k <= IC s ; :: thesis: (IC (DecIC (s,k))) + k = IC s
thus (IC (DecIC (s,k))) + k = ((IC s) -' k) + k by Th65
.= IC s by A1, XREAL_1:235 ; :: thesis: verum