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 Th66:
  F is associative & F is having_a_unity & F is having_an_inverseOp &
  G is_distributive_wrt F & e = the_unity_wrt F implies
  for d holds G.(e,d) = e & G.(d,e) = e
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 and
A5: e = the_unity_wrt F;
  let d;
  set ed = G.(e,d);
  F.(ed,ed) = G.(F.(e,e),d) by A4,BINOP_1:11
    .= ed by A2,A5,SETWISEO:15;
  hence ed = e by A1,A2,A3,A5,Th65;
  set de = G.(d,e);
  F.(de,de) = G.(d,F.(e,e)) by A4,BINOP_1:11
    .= de by A2,A5,SETWISEO:15;
  hence thesis by A1,A2,A3,A5,Th65;
end;
