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 Th34:
  F is commutative associative & F is having_a_unity & len p = len
  q implies F.(F"**"p,F"**"q) = F "**"(F.:(p,q))
proof
  set e = the_unity_wrt F;
  assume
A1: F is commutative & F is associative & F is having_a_unity;
  then
  F.(e,e) = e & for d1,d2,d3,d4 holds F.(F.(d1,d2),F.(d3,d4))= F.(F.(d1,d3
  ),F. (d2,d4)) by Lm3,SETWISEO:15;
  hence thesis by A1,Th32;
end;
