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;

theorem Th53:
   Loc-Ideal(p) is proper Ideal of A~p
   proof
     reconsider M = Loc-Ideal(p) as Subset of A~p;
A1:  for m, n being Element of A~p st m in M & n in M holds m+n in M
     proof
       let m, n being Element of A~p;
       assume that
A2:    m in M and
A3:    n in M;
       consider y1 be Element of A~p such that
A4:    y1 = m and
A5:    ex a be Element of Frac(Loc(A,p))
       st a in [:p,Loc(A,p):] & y1 = Class(EqRel(Loc(A,p)),a) by A2;
       consider y2 be Element of A~p such that
A6:    y2 = n and
A7:    ex a be Element of Frac(Loc(A,p))
       st a in [:p,Loc(A,p):] & y2 = Class(EqRel(Loc(A,p)),a) by A3;
       consider a1 be Element of Frac(Loc(A,p)) such that
A8:    a1 in [:p,Loc(A,p):] and
A9:    y1 = Class(EqRel(Loc(A,p)),a1) by A5;
       consider a2 be Element of Frac(Loc(A,p)) such that
A10:   a2 in [:p,Loc(A,p):] and
A11:   y2 = Class(EqRel(Loc(A,p)),a2) by A7;
A12:   a1`1 in p & a2`1 in p by A8,A10,MCART_1:10;
       a1`2 in Loc(A,p) & a2`2 in Loc(A,p) by A8,A10,MCART_1:10; then
A17:   a1`2 * a2`2 in Loc(A,p) by C0SP1:def 4;
A14:   p is prime Ideal of A by Lm5; then
A15:   a1`1 * a2`2 in p by A12,IDEAL_1:def 3;
       a2`1 * a1`2 in p by A12,A14,IDEAL_1:def 2; then
A16:   a1`1 * a2`2 + a2`1 * a1`2 in p by A14,A15,IDEAL_1:def 1;
       reconsider a3 = a1 + a2 as Element of Frac(Loc(A,p));
A18:   a3 in [:p,Loc(A,p):] by A16,A17,ZFMISC_1:87;
       reconsider y3 = y1 + y2 as Element of A~p;
       y3 = Class(EqRel(Loc(A,p)),a3) by A9,A11,Th35;
       hence thesis by A4,A6,A18;
     end;
     for x,m being Element of A~p st m in M holds x*m in M
     proof
       let x,m be Element of A~p;
       assume m in M; then
       consider y1 be Element of A~p such that
A20:   y1 = m and
A21:   ex a be Element of Frac(Loc(A,p))
       st a in [:p,Loc(A,p):] & y1 = Class(EqRel(Loc(A,p)),a);
       consider a1 be Element of Frac(Loc(A,p)) such that
A22:   a1 in [:p,Loc(A,p):] and
A23:   y1 = Class(EqRel(Loc(A,p)),a1) by A21;
       consider b being Element of Frac(Loc(A,p)) such that
A24:   x = Class(EqRel(Loc(A,p)),b) by Th32;
A25:   a1`1 in p & a1`2 in Loc(A,p) by A22,MCART_1:10;
       b in Frac(Loc(A,p)); then
       b in [:[#]A, Loc(A,p):] by Th15; then
       b`1 in [#]A & b`2 in Loc(A,p) by MCART_1:10; then
A28:   b`2 * a1`2 in Loc(A,p) by A25,C0SP1:def 4;
       reconsider ba1 = b*a1 as Element of Frac(Loc(A,p));
       p is prime Ideal of A by Lm5; then
       b`1 * a1`1 in p by A22,MCART_1:10,IDEAL_1:def 2; then
A29:   ba1 in [:p,Loc(A,p):] by A28,ZFMISC_1:87;
       reconsider xy = x*y1 as Element of A~p;
       xy = Class(EqRel(Loc(A,p)),ba1) by A23,A24,Th33;
       hence thesis by A20,A29;
     end; then
A31: M is left-ideal by IDEAL_1:def 2;
     M is proper
     proof
       assume not M is proper; then
       1.(A~p) in M by A31,IDEAL_1:19; then
       consider y1 be Element of A~p such that
A34:   y1 = 1.(A~p) and
A35:   ex a be Element of Frac(Loc(A,p))
       st a in [:p,Loc(A,p):] & y1 = Class(EqRel(Loc(A,p)),a);
       consider a be Element of Frac(Loc(A,p)) such that
A36:   a in [:p,Loc(A,p):] and
A37:   y1 = Class(EqRel(Loc(A,p)),a) by A35;
A38:   (frac1(Loc(A,p))).(1.A) = [1.A,1.A] by Def4;
       Class(EqRel(Loc(A,p)),a)
       = Class(EqRel(Loc(A,p)),(frac1(Loc(A,p))).(1.A)) by A34,A37,Lm43; then
A39:   a, (frac1(Loc(A,p))).(1.A) Fr_Eq Loc(A,p) by Th26;
       reconsider y = (frac1(Loc(A,p))).(1.A) as Element of Frac(Loc(A,p));
       consider s1 be Element of A such that
A40:  s1 in Loc(A,p) and
A41:  (a`1 * y`2 - y`1 * a`2) * s1 = 0.A by A39;
      0.A = a`1*s1 - a`2 * s1 by A38,A41,VECTSP_1:13; then
A42:  a`1*s1 = a`2 * s1 by VECTSP_1:27;
A43:  a`1 in p & a`2 in Loc(A,p) by A36,MCART_1:10;
      p is prime Ideal of A by Lm5; then
A44:  a`1 * s1 in p by A36,MCART_1:10,IDEAL_1:def 2;
      a`2 * s1 in Loc(A,p) by A40,A43,C0SP1:def 4;
      hence contradiction by A42,A44,XBOOLE_0:def 5;
     end;
     hence thesis by A1,A31,IDEAL_1:def 1;
   end;
