let N be with_non-empty_elements set ; :: thesis: for S being non empty IC-Ins-separated Mem-Struct of 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 IC-Ins-separated Mem-Struct of 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 Z: k <= IC s ; :: thesis: (IC (DecIC (s,k))) + k = IC s
thus (IC (DecIC (s,k))) + k = ((IC s) -' k) + k by Th188
.= IC s by Z, XREAL_1:235 ; :: thesis: verum