theorem
  H is commutative associative & (B <> {} or H is having_a_unity) & f is
  one-to-one implies H $$(f.:B,h) = H $$(B,h*f)
proof
  assume that
A1: H is commutative associative &( B <> {} or H is having_a_unity) and
A2: f is one-to-one;
  set s = f|B;
A3: rng s = f.:B & (h*f)|B = h*s by RELAT_1:83,115;
  B c= C by FINSUB_1:def 5;
  then B c= dom f by FUNCT_2:def 1;
  then
A4: dom s = B by RELAT_1:62;
  s is one-to-one by A2,FUNCT_1:52;
  hence thesis by A1,A4,A3,Th5;
end;
