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 Th63:
  F is having_a_unity & F is associative & F is commutative &
  F is having_an_inverseOp implies
    (the_inverseOp_wrt F) is_distributive_wrt F
proof
  assume that
A1: F is having_a_unity and
A2: F is associative and
A3: F is commutative and
A4: F is having_an_inverseOp;
  let d1,d2;
  set e = the_unity_wrt F, u = the_inverseOp_wrt F;
  F.(F.(u.d1,u.d2),F.(d1,d2)) = F.(u.d1,F.(u.d2,F.(d1,d2))) by A2
    .= F.(u.d1,F.(F.(u.d2,d1),d2)) by A2
    .= F.(u.d1,F.(F.(d1,u.d2),d2)) by A3
    .= F.(u.d1,F.(d1,F.(u.d2,d2))) by A2
    .= F.(u.d1,F.(d1,e)) by A1,A2,A4,Th59
    .= F.(F.(u.d1,d1),e) by A2
    .= F.(e,e) by A1,A2,A4,Th59
    .= e by A1,SETWISEO:15;
  hence u.(F.(d1,d2)) = F.(u.d1,u.d2) by A1,A2,A4,Th60;
end;
