reserve x,y for set;
reserve C,C9,D,D9,E for non empty set;
reserve c for Element of C;
reserve c9 for Element of C9;
reserve d,d1,d2,d3,d4,e for Element of D;
reserve d9 for Element of D9;
reserve i,j for natural Number;
reserve F for Function of [:D,D9:],E;
reserve p,q for FinSequence of D,
  p9,q9 for FinSequence of D9;
reserve f,f9 for Function of C,D,
  h for Function of D,E;
reserve T,T1,T2,T3 for Tuple of i,D;
reserve T9 for Tuple of i, D9;
reserve S for Tuple of j, D;
reserve S9 for Tuple of j, D9;
reserve F,G for BinOp of D;
reserve u for UnOp of D;
reserve H for BinOp of E;

theorem
  F is associative & F is having_a_unity & F is having_an_inverseOp &
  G is_distributive_wrt F implies
    G[;](d,id D).the_unity_wrt F = the_unity_wrt F
proof
  assume that
A1: F is associative and
A2: F is having_a_unity and
A3: F is having_an_inverseOp and
A4: G is_distributive_wrt F;
  set e = the_unity_wrt F, i = the_inverseOp_wrt F;
  G.(d,e) = G.(d,F.(e,e)) by A2,SETWISEO:15
    .= F.(G.(d,e),G.(d,e)) by A4,BINOP_1:11;
  then e = F.(F.(G.(d,e),G.(d,e)),i.(G.(d,e))) by A1,A2,A3,Th59;
  then e = F.(G.(d,e),F.(G.(d,e),i.(G.(d,e)))) by A1;
  then e = F.(G.(d,e),e) by A1,A2,A3,Th59;
  then e = G.(d,e) by A2,SETWISEO:15;
  then G.(d,(id D).e) = e;
  hence thesis by FUNCOP_1:53;
end;
