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
  F is commutative associative & (B <> {} or F is having_a_unity) & f|B
  = f9|B implies F $$(B,f) = F $$(B,f9)
proof
  assume
A1: F is commutative associative &( B <> {} or F is having_a_unity);
  set s = id B;
A2: dom s = B & rng s = B;
  assume f|B = f9|B;
  then f|B = f9*s by RELAT_1:65;
  hence thesis by A1,A2,Th5;
end;
