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 Th60:
  F is having_a_unity & F is associative & F is having_an_inverseOp &
  F.(d1,d2) = the_unity_wrt F implies
    d1 = (the_inverseOp_wrt F).d2 & (the_inverseOp_wrt F).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.(d1,d2) = the_unity_wrt F;
  set e = the_unity_wrt F, d3 = (the_inverseOp_wrt F).d2;
  F.(F.(d1,d2),d3) = d3 by A1,A4,SETWISEO:15;
  then F.(d1,F.(d2,d3)) = d3 by A2;
  then F.(d1,e) = d3 by A1,A2,A3,Th59;
  hence d1 = d3 by A1,SETWISEO:15;
  set d3 = (the_inverseOp_wrt F).d1;
  F.(d3,F.(d1,d2)) = d3 by A1,A4,SETWISEO:15;
  then F.(F.(d3,d1),d2) = d3 by A2;
  then F.(e,d2) = d3 by A1,A2,A3,Th59;
  hence thesis by A1,SETWISEO:15;
end;
