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 Th64:
  F is having_a_unity & F is associative & F is having_an_inverseOp &
  (F.(d,d1) = F.(d,d2) or F.(d1,d) = F.(d2,d)) implies d1 = d2
proof
  assume that
A1: F is having_a_unity and
A2: F is associative and
A3: F is having_an_inverseOp and
A4: F.(d,d1) = F.(d,d2) or F.(d1,d) = F.(d2,d);
  set e = the_unity_wrt F, u = the_inverseOp_wrt F;
  per cases by A4;
  suppose
A5: F.(d,d1) = F.(d,d2);
    thus d1 = F.(e,d1) by A1,SETWISEO:15
      .= F.(F.(u.d,d),d1) by A1,A2,A3,Th59
      .= F.(u.d,F.(d,d2)) by A2,A5
      .= F.(F.(u.d,d),d2) by A2
      .= F.(e,d2) by A1,A2,A3,Th59
      .= d2 by A1,SETWISEO:15;
  end;
  suppose
A6: F.(d1,d) = F.(d2,d);
    thus d1 = F.(d1,e) by A1,SETWISEO:15
      .= F.(d1,F.(d,u.d)) by A1,A2,A3,Th59
      .= F.(F.(d2,d),u.d) by A2,A6
      .= F.(d2,F.(d,u.d)) by A2
      .= F.(d2,e) by A1,A2,A3,Th59
      .= d2 by A1,SETWISEO:15;
  end;
end;
