reserve R,R1 for commutative Ring;
reserve A,B for non degenerated commutative Ring;
reserve o,o1,o2 for object;
reserve r,r1,r2 for Element of R;
reserve a,a1,a2,b,b1 for Element of A;
reserve f for Function of R, R1;
reserve p for Element of Spectrum A;
reserve S for non empty multiplicatively-closed Subset of R;
reserve u,v,w,x,y,z for Element of Frac(S);
reserve a, b, c for Element of Frac(S);
reserve x, y, z for Element of S~R;
reserve S for without_zero non empty multiplicatively-closed Subset of A;
reserve p for Element of Spectrum A;
reserve a,m,n for Element of A~p;
reserve f for Function of A,B;
reserve x for object;
reserve A for domRing;

theorem Th65:
  Total-Quotient-Ring(A) is Field
  proof
    set S = Non_ZeroDiv_Set(A);
A0: S = [#]A \ {0.A} by Th4;
    for x be Element of S~A st x <> 0.(S~A) holds
    x is left_invertible
    proof
      let x be Element of S~A;
      assume
A1:   x <> 0.(S~A);
      consider a1,s1 being Element of A such that
A2:   s1 in S and
A3:   x = Class(EqRel(S),[a1,s1]) by Th46;
      reconsider as1 = [a1,s1] as Element of Frac(S) by A2,Def3;
      a1 <> 0.A
      proof
        assume
A5:     a1 = 0.A;
        reconsider t = 1.A as Element of A;
        t in S by C0SP1:def 4; then
        as1, 0.(A,S) Fr_Eq S by A5; then
        x = Class(EqRel(S),0.(A,S)) by A3,Th26 .= 0.(S~A) by Def6;
        hence contradiction by A1;
      end; then
      not a1 in {0.A} by TARSKI:def 1; then
      as1`1 in S by A0,XBOOLE_0:def 5; then
      Class(EqRel(S),as1) is Unit of S~A by Lm45; then
      x in Unit_Set(S~A) by A3; then
      x["]*x = 1.(S~A) by Def2;
      hence thesis;
    end;
    then S~A is almost_left_invertible;
    hence thesis;
  end;
