reserve x,y for set;
reserve C,C9,D,E for non empty set;
reserve c,c9,c1,c2,c3 for Element of C;
reserve B,B9,B1,B2 for Element of Fin C;
reserve A for Element of Fin C9;
reserve d,d1,d2,d3,d4,e for Element of D;
reserve F,G for BinOp of D;
reserve u for UnOp of D;
reserve f,f9 for Function of C,D;
reserve g for Function of C9,D;
reserve H for BinOp of E;
reserve h for Function of D,E;
reserve i,j for Nat;
reserve s for Function;
reserve p,q for FinSequence of D;
reserve T1,T2 for Element of i-tuples_on D;

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;
