let il be Nat; :: thesis: for S being COM-Struct
for k being Nat
for P being preProgram of S holds
( il in dom P iff il + k in dom (Reloc (P,k)) )
let S be COM-Struct ; :: thesis: for k being Nat
for P being preProgram of S holds
( il in dom P iff il + k in dom (Reloc (P,k)) )
let k be Nat; :: thesis: for P being preProgram of S holds
( il in dom P iff il + k in dom (Reloc (P,k)) )
let P be preProgram of S; :: thesis: ( il in dom P iff il + k in dom (Reloc (P,k)) )
A1: dom (Reloc (P,k)) = { (j + k) where j is Nat : j in dom P } by Th21;
reconsider il1 = il as Element of NAT by ORDINAL1:def 12;
( il1 in dom P implies il1 + k in dom (Reloc (P,k)) ) by A1;
hence ( il in dom P implies il + k in dom (Reloc (P,k)) ) ; :: thesis: ( il + k in dom (Reloc (P,k)) implies il in dom P )
assume il + k in dom (Reloc (P,k)) ; :: thesis: il in dom P
then ex j being Nat st
( il + k = j + k & j in dom P ) by A1;
hence il in dom P ; :: thesis: verum