theorem Th67:
  F is having_a_unity & F is associative & F is having_an_inverseOp &
  u = the_inverseOp_wrt F & G is_distributive_wrt F implies
  u.(G.(d1,d2)) = G.(u.d1,d2) & u.(G.(d1,d2)) = G.(d1,u.d2)
proof
  assume that
A1: F is having_a_unity & F is associative & F is having_an_inverseOp and
A2: u = the_inverseOp_wrt F and
A3: G is_distributive_wrt F;
  set e = the_unity_wrt F;
  F.(G.(d1,d2),G.(u.d1,d2)) = G.(F.(d1,u.d1),d2) by A3,BINOP_1:11
    .= G.(e,d2) by A1,A2,Th59
    .= e by A1,A3,Th66;
  hence u.(G.(d1,d2)) = G.(u.d1,d2) by A1,A2,Th60;
  F.(G.(d1,d2),G.(d1,u.d2)) = G.(d1,F.(d2,u.d2)) by A3,BINOP_1:11
    .= G.(d1,e) by A1,A2,Th59
    .= e by A1,A3,Th66;
  hence thesis by A1,A2,Th60;
end;
