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 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;
