let A, B, C be Ordinal; :: thesis: ( A c= B implies A *^ C c= B *^ C )
assume A1: A c= B ; :: thesis: A *^ C c= B *^ C
A2: now :: thesis: ( C <> 0 implies A *^ C c= B *^ C )
assume A3: C <> 0 ; :: thesis: A *^ C c= B *^ C
now :: thesis: ( A <> B implies A *^ C c= B *^ C )
assume A <> B ; :: thesis: A *^ C c= B *^ C
then A c< B by A1;
then A *^ C in B *^ C by A3, Th40, ORDINAL1:11;
hence A *^ C c= B *^ C by ORDINAL1:def 2; :: thesis: verum
end;
hence A *^ C c= B *^ C ; :: thesis: verum
end;
( C = 0 implies A *^ C c= B *^ C ) by Th38;
hence A *^ C c= B *^ C by A2; :: thesis: verum