 reserve o for object;
 reserve F for non almost_trivial Field;
 reserve x,a for Element of F;

theorem Th10:
   for x being non trivial Element of F, o being object st not o in [#]F
   holds ExField(x,o) is associative
   proof
    let x be non trivial Element of F;
    let o be object;
    assume not o in [#]F; then
A1: a <> o;
set C = carr(x,o), E = ExField(x,o);
set MULTR = the multF of F;
A2: [#]E = C by Def8;
    now let a,b,c be Element of E;
     per cases;
      suppose
A3:    a = o; then
       a in {o} by TARSKI:def 1; then
       reconsider a1 = a as Element of C by XBOOLE_0:def 3;
       per cases;
        suppose
A4:      b = o; then
         b in {o} by TARSKI:def 1; then
         reconsider b1 = b as Element of C by XBOOLE_0:def 3;
         per cases;
          suppose
A5:        (MULTR).(x,x) <> x;
A6:        a * b = (multR(x,o)).(a1,b1) by Def8 .= multR(a1,b1) by Def7
           .= x * x by A5,A4,A3,Def6;
           not x * x in {x} by A5,TARSKI:def 1; then
           x * x in [#]F \ {x} by XBOOLE_0:def 5; then
           reconsider xx = x * x as Element of C by XBOOLE_0:def 3;
A7:        xx <> o by A1;
           per cases;
            suppose
A8:          c = o; then
             c in {o} by TARSKI:def 1; then
             reconsider c1 = c as Element of C by XBOOLE_0:def 3;
A9:          (a * b) * c = (multR(x,o)).(a*b,c1) by Def8
             .= multR(xx,c1) by A6,Def7;
A10:         b * c = (multR(x,o)).(b1,c1) by Def8
             .= multR(b1,c1) by Def7 .= x * x by A4,A5,A8,Def6;
A11:         (MULTR).(x,xx) = x * (x * x)
             .= (x * x) * x by GROUP_1:def 3;
             per cases;
              suppose
A12:           (MULTR).(xx,x) <> x;
               hence (a * b) * c = (x * x) * x by A8,A1,A9,Def6
               .= multR(a1,xx) by A3,A11,A12,A1,Def6
               .= (multR(x,o)).(a1,xx) by Def7
               .= a * (b * c) by A10,Def8;
              end;
              suppose
A13:           (MULTR).(xx,x) = x;
               hence (a * b) * c = o by A9,A7,A8,Def6
               .= multR(a1,xx) by A7,A11,A13,A3,Def6
               .= (multR(x,o)).(a1,xx) by Def7
               .= a * (b * c) by A10,Def8;
              end;
             end;
             suppose
A14:          c <> o; then
              not c in {o} by TARSKI:def 1; then
              c in [#]F \ {x} by A2,XBOOLE_0:def 3; then
              reconsider cR = c as Element of F;
              reconsider c1 = c as Element of C by Def8;
A15:          (a * b) * c = (multR(x,o)).(a*b,c1) by Def8
              .= multR(xx,c1) by A6,Def7;
A16:          b * c = (multR(x,o)).(b1,c1) by Def8
              .= multR(b1,c1) by Def7; then
              reconsider bc = b * c as Element of C;
              per cases;
               suppose
A17:            (MULTR).(x,c1) <> x; then
A18:            b * c = x * cR by A16,A14,A4,Def6; then
A19:            b * c <> o by A1;
                per cases;
                 suppose (MULTR).(xx,c1) <> x; then
A20:              (a * b) * c = (x * x) * cR by A7,A15,A14,Def6
                  .= x * (x * cR) by GROUP_1:def 3
                  .= (MULTR).(x,b*c) by A17,A16,A14,A4,Def6;
                  per cases;
                   suppose (MULTR).(x,bc) <> x;
                    hence (a * b) * c = multR(a1,bc) by A18,A1,A3,Def6,A20
                    .= (multR(x,o)).(a1,b*c) by Def7
                    .= a * (b * c) by Def8;
                   end;
                   suppose
A21:                (MULTR).(x,bc) = x;
A22:                (MULTR).(x,bc) = x * (x * cR) by A17,A16,A14,A4,Def6
                    .= (x * x) * cR by GROUP_1:def 3
                    .= (MULTR).(xx,c1);
                    thus
                    (a * b) * c = o by A15,A7,A14,A22,A21,Def6
                    .= multR(a1,bc) by A3,A19,A21,Def6
                    .= (multR(x,o)).(a1,b*c) by Def7
                    .= a * (b * c) by Def8;
                   end;
                  end;
                  suppose
A23:               (MULTR).(xx,c1) = x; then
A24:               (a * b) * c = o by A7,A15,A14,Def6;
                   per cases;
                    suppose (MULTR).(x,bc) <> x;
A25:                 (MULTR).(x,bc) = x * (x * cR) by A17,A16,A14,A4,Def6
                     .= (x * x) * cR by GROUP_1:def 3
                     .= (MULTR).(xx,c1);
                     thus
                     (a * b) * c = multR(a1,bc) by A25,A23,A19,A3,Def6,A24
                     .= (multR(x,o)).(a1,b*c) by Def7 .= a * (b * c) by Def8;
                    end;
                    suppose
A26:                 (MULTR).(x,bc) = x;
A27:                 (MULTR).(x,bc) = x * (x * cR) by A17,A16,A14,A4,Def6
                     .= (x * x) * cR by GROUP_1:def 3
                     .= (MULTR).(xx,c1);
                     thus
                     (a * b) * c = o by A15,A7,A14,A27,A26,Def6
                     .= multR(a1,bc) by A19,A26,A3,Def6
                     .= (multR(x,o)).(a1,b*c) by Def7 .= a * (b * c) by Def8;
                    end;
                   end;
                  end;
                  suppose
A28:               (MULTR).(x,c1) = x;
A29:               x <> 0.F by Def2;
A30:               x = x * cR  by A28 .= cR * x by GROUP_1:def 12;
A31:               cR = cR * 1_F .= cR * (x * x") by A29,VECTSP_2:9
                   .= x * x" by A30,GROUP_1:def 3 .= 1_F by A29,VECTSP_2:9;
A32:               b * c = o by A28,A16,A14,A4,Def6;
                   per cases;
                    suppose (MULTR).(xx,c1) <> x;
                     hence (a * b) * c = (x * x) * cR by A15,A7,A14,Def6
                     .= multR(a1,bc) by A31,A32,A5,A3,Def6
                     .= (multR(x,o)).(a1,b*c) by Def7 .= a * (b * c) by Def8;
                    end;
                    suppose (MULTR).(xx,c1) = x; then
                     x = (x * x) * cR .= x * x by A31;
                     hence (a * b) * c = a * (b * c) by A5;
                    end;
                  end;
                end;
              end;
              suppose
A34:           (MULTR).(x,x) = x;
A35:           a * b = (multR(x,o)).(a1,b1) by Def8 .= multR(a1,b1) by Def7
               .= o by A34,A4,A3,Def6; then
               a * b in {o} by TARSKI:def 1; then
               reconsider ab = a * b as Element of C by XBOOLE_0:def 3;
               per cases;
                suppose
A36:             c = o; then
                 c in {o} by TARSKI:def 1; then
                 reconsider c1 = c as Element of C by XBOOLE_0:def 3;
A37:             b * c = (multR(x,o)).(b1,c1) by Def8
                 .= multR(b1,c1) by Def7 .= o by A34,A36,A4,Def6; then
                 b * c in {o} by TARSKI:def 1; then
                 reconsider bc = b * c as Element of C by XBOOLE_0:def 3;
                 thus
                 (a * b) * c = a * (b * c) by A35,A36,A37,A3;
                end;
                suppose
A38:             c <> o; then
                 not c in {o} by TARSKI:def 1; then
                 c in [#]F \ {x} by A2,XBOOLE_0:def 3; then
                 reconsider cR = c as Element of F;
                 reconsider c1 = c as Element of C by Def8;
                 per cases;
                  suppose
A39:               (MULTR).(x,c1) = x;
A40:               b * c = (multR(x,o)).(b1,c1) by Def8
                   .= multR(b1,c1) by Def7 .= o by A39,A38,A4,Def6; then
                   b * c in {o} by TARSKI:def 1; then
                   reconsider bc = b * c as Element of C by XBOOLE_0:def 3;
                   thus
                   (a * b) * c = (multR(x,o)).(ab,c1) by Def8
                   .= multR(ab,c1) by Def7 .= o by A39,A35,A38,Def6
                   .= multR(a1,bc) by A40,A34,A3,Def6
                   .= (multR(x,o)).(a1,b*c) by Def7 .= a * (b * c) by Def8;
                  end;
                  suppose
A41:               (MULTR).(x,c1) <> x;
A42:               b * c = (multR(x,o)).(b1,c1) by Def8.= multR(b1,c1) by Def7
                   .= x * cR by A41,A38,A4,Def6; then
                   b * c <> o by A1; then
                   not b * c in {o} by TARSKI:def 1; then
                   b * c in [#]F \ {x} by A2,XBOOLE_0:def 3; then
                   reconsider bcR = b * c as Element of F;
                   reconsider bc = b*c as Element of C by Def8;
A43:               (a * b) * c  = (multR(x,o)).(ab,c1) by Def8
                   .= multR(ab,c1) by Def7
                   .= (MULTR).(x,c1) by A35,A38,A41,Def6;
                   (a * b) * c = (multR(x,o)).(ab,c1) by Def8
                   .= multR(ab,c1) by Def7
                   .= (x * x) * cR by A34,A35,A38,A41,Def6
                   .= x * (x * cR) by GROUP_1:def 3 .= (MULTR).(x,bc) by A42;
                   hence
                   (a * b) * c = multR(a1,bc) by A41,A43,A42,A1,A3,Def6
                   .= (multR(x,o)).(a1,b*c) by Def7 .= a * (b * c) by Def8;
                  end;
                end;
               end;
             end;
             suppose
A44:          b <> o; then
              not b in {o} by TARSKI:def 1; then
              b in [#]F \ {x} by A2,XBOOLE_0:def 3; then
              reconsider bR = b as Element of F;
              reconsider b1 = b as Element of C by Def8;
A45:          (MULTR).(x,b) = x * bR .= bR * x by GROUP_1:def 12
              .= (MULTR).(b,x);
              per cases;
               suppose
A46:            (MULTR).(x,b) <> x;
A47:            a * b = (multR(x,o)).(a1,b1) by Def8 .= multR(a1,b1) by Def7
                .= x * bR by A46,A44,A3,Def6; then
A48:            a * b <> o by A1; then
                not a * b in {o} by TARSKI:def 1; then
                a * b in [#]F \ {x} by A2,XBOOLE_0:def 3; then
                reconsider abR = a * b as Element of F;
                reconsider ab = a*b as Element of C by Def8;
                per cases;
                 suppose
A49:              c = o; then
                  c in {o} by TARSKI:def 1; then
                  reconsider c1 = c as Element of C by XBOOLE_0:def 3;
A50:              b * c = (multR(x,o)).(b1,c1) by Def8
                  .= multR(b1,c1) by Def7
                  .= bR * x by A45,A46,A49,A44,Def6; then
A51:              b * c <> o by A1; then
                  not b * c in {o} by TARSKI:def 1; then
                  b * c in [#]F \ {x} by A2,XBOOLE_0:def 3;then
                  reconsider bcR = b * c as Element of F;
                  reconsider bc = b*c as Element of C by Def8;
A52:              (MULTR).(ab,x) = (x * bR) * x by A47
                  .= x * (bR * x) by GROUP_1:def 3;
                  per cases;
                   suppose
A53:                (MULTR).(ab,x) <> x;
                    thus (a * b) * c = (multR(x,o)).(ab,c1) by Def8
                    .= multR(ab,c1) by Def7
                    .= (MULTR).(ab,x) by A47,A1,Def6,A49,A53
                    .= multR(a1,bc) by A53,A52,A50,A1,A3,Def6
                    .= (multR(x,o)).(a,bc) by Def7 .= a * (b * c) by Def8;
                   end;
                   suppose
A54:                (MULTR).(ab,x) = x;
                    thus (a * b) * c = (multR(x,o)).(ab,c1) by Def8
                    .= multR(ab,c1) by Def7 .= o by A48,A49,A54,Def6
                    .= multR(a1,bc) by A54,A52,A50,A51,A3,Def6
                    .= (multR(x,o)).(a,bc) by Def7 .= a * (b * c) by Def8;
                   end;
                 end;
                 suppose
A55:              c <> o; then
                  not c in {o} by TARSKI:def 1; then
                  c in [#]F \ {x} by A2,XBOOLE_0:def 3; then
                  reconsider cR = c as Element of F;
                  reconsider c1 = c as Element of C by Def8;
A56:              (MULTR).(ab,c) = (x * bR) * cR by A47
                  .= x * (bR * cR) by GROUP_1:def 3;
                  per cases;
                  suppose
A57:               (MULTR).(b,c) <> x;
A58:               b * c = (multR(x,o)).(b1,c1) by Def8
                   .= multR(b1,c1) by Def7
                   .= bR * cR by A57,A55,A44,Def6; then
A59:               b * c <> o by A1; then
                   not b * c in {o} by TARSKI:def 1; then
                   b * c in [#]F \ {x} by A2,XBOOLE_0:def 3; then
                   reconsider bcR = b * c as Element of F;
                   reconsider bc = b*c as Element of C by Def8;
                   per cases;
                   suppose
A60:                (MULTR).(ab,c1) <> x;
                    thus (a * b) * c = (multR(x,o)).(ab,c1) by Def8
                    .= multR(ab,c1) by Def7
                    .= (x * bR) * cR by A47,A60,A55,A48,Def6
                    .= x * (bR * cR) by GROUP_1:def 3
                    .= multR(a1,bc) by A3,A56,A60,A58,A1,Def6
                    .= (multR(x,o)).(a,bc) by Def7 .= a * (b * c) by Def8;
                   end;
                   suppose
A61:                (MULTR).(ab,c1) = x;
                    thus (a * b) * c = (multR(x,o)).(ab,c1) by Def8
                    .= multR(ab,c1) by Def7 .= o by A61,A55,A48,Def6
                    .= multR(a1,bc) by A3,A56,A61,A58,A59,Def6
                    .= (multR(x,o)).(a,bc) by Def7 .= a * (b * c) by Def8;
                   end;
                 end;
                 suppose
A62:              (MULTR).(b,c) = x;
A63:              b * c = (multR(x,o)).(b1,c1) by Def8 .= multR(b1,c1) by Def7
                  .= o by A62,A55,A44,Def6; then
                  b * c in {o} by TARSKI:def 1; then
                  reconsider bc = b * c as Element of C by XBOOLE_0:def 3;
                  per cases;
                   suppose
A64:                (MULTR).(ab,c1) <> x;
                    thus (a * b) * c = (multR(x,o)).(ab,c1) by Def8
                    .= multR(ab,c1) by Def7
                    .= (x * bR) * cR by A47,A64,A55,A48,Def6
                    .= x * (bR * cR) by GROUP_1:def 3
                    .= multR(a1,bc) by A62,A3,A56,A64,A63,Def6
                    .= (multR(x,o)).(a,bc) by Def7 .= a * (b * c) by Def8;
                   end;
                   suppose
A65:                (MULTR).(ab,c1) = x;
                    thus (a * b) * c = (multR(x,o)).(ab,c1) by Def8
                    .= multR(ab,c1) by Def7 .= o by A65,A55,A48,Def6
                    .= multR(a1,bc) by A62,A3,A56,A65,A63,Def6
                    .= (multR(x,o)).(a,bc) by Def7 .= a * (b * c) by Def8;
                   end;
                  end;
                 end;
               end;
               suppose
A66:            (MULTR).(x,b) = x;
A67:            x <> 0.F by Def2;
A68:            x = x * bR by A66 .= bR * x by GROUP_1:def 12;
A69:            bR = bR * 1_F .= bR * (x * x") by A67,VECTSP_2:9
                .= x * x" by A68,GROUP_1:def 3 .= 1_F by A67,VECTSP_2:9;
A70:            a * b = (multR(x,o)).(a1,b1) by Def8 .= multR(a1,b1) by Def7
                .= o by A66,A44,A3,Def6; then
                a * b in {o} by TARSKI:def 1; then
                reconsider ab = a * b as Element of C by XBOOLE_0:def 3;
                per cases;
                 suppose
A71:             c = o; then
                 c in {o} by TARSKI:def 1; then
                 reconsider c1 = c as Element of C by XBOOLE_0:def 3;
A72:             b * c = (multR(x,o)).(b1,c1) by Def8 .= multR(b1,c1) by Def7
                 .= o by A45,A66,A71,A44,Def6; then
                 b * c in {o} by TARSKI:def 1; then
                 reconsider bc = b * c as Element of C by XBOOLE_0:def 3;
                 thus
                 (a * b) * c = a * (b * c) by A70,A71,A72,A3;
                end;
                suppose
A73:             c <> o; then
                 not c in {o} by TARSKI:def 1; then
A74:             c in [#]F \ {x} by A2,XBOOLE_0:def 3; then
                 reconsider cR = c as Element of F;
                 reconsider c1 = c as Element of C by Def8;
A75:             bR <> 0.F by A69;
A76:             now assume (MULTR).(b,c) = x; then
                  bR * cR = x * bR by A66 .= bR * x by GROUP_1:def 12; then
                  cR = x by A75,ALGSTR_0:def 36,ALGSTR_0:def 20; then
                  c in {x} by TARSKI:def 1;
                  hence contradiction by A74,XBOOLE_0:def 5;
                 end;
A78:             b * c = (multR(x,o)).(b1,c1) by Def8
                 .= multR(b1,c1) by Def7 .= bR * cR by A76,A73,A44,Def6; then
A79:             b * c <> o by A1; then
                 not b * c in {o} by TARSKI:def 1; then
                 b * c in [#]F \ {x} by A2,XBOOLE_0:def 3; then
                 reconsider bcR = b * c as Element of F;
                 reconsider bc = b*c as Element of C by Def8;
A80:             x * (bR * cR) = (x * bR) * cR by GROUP_1:def 3
                 .= x * cR by A66;
                 per cases;
                  suppose
A81:              (MULTR).(x,c1) <> x;
                  thus (a * b) * c = (multR(x,o)).(ab,c1) by Def8
                  .= multR(ab,c1) by Def7
                  .= (MULTR).(x,c1) by A70,A81,A73,Def6
                  .= multR(a1,bc) by A80,A3,A81,A78,A1,Def6
                  .= (multR(x,o)).(a,bc) by Def7 .= a * (b * c) by Def8;
                 end;
                 suppose
A82:              (MULTR).(x,c1) = x;
                  thus (a * b) * c = (multR(x,o)).(ab,c1) by Def8
                  .= multR(ab,c1) by Def7 .= o by A82,A73,A70,Def6
                  .= multR(a1,bc) by A79,A82,A80,A3,A78,Def6
                  .= (multR(x,o)).(a,bc) by Def7 .= a * (b * c) by Def8;
                 end;
                end;
               end;
              end;
             end;
       suppose
A83:    a <> o; then
        not a in {o} by TARSKI:def 1; then
A84:    a in [#]F \ {x} by A2,XBOOLE_0:def 3; then
        reconsider aR = a as Element of F;
        reconsider a1 = a as Element of C by Def8;
        per cases;
         suppose
A85:      b = o; then
          b in {o} by TARSKI:def 1; then
          reconsider b1 = b as Element of C by XBOOLE_0:def 3;
          per cases;
           suppose
A86:        (MULTR).(a1,x) = x;
A87:        x <> 0.F by Def2;
            aR = aR * 1_F .= aR * (x * x") by A87,VECTSP_2:9
            .= (aR * x) * x" by GROUP_1:def 3
            .= 1_F by A86,A87,VECTSP_2:9; then
A88:        aR <> 0.F;
A89:        a * b = (multR(x,o)).(a1,b1) by Def8 .= multR(a1,b1) by Def7
            .= o by A86,A85,A83,Def6; then
            a * b in {o} by TARSKI:def 1; then
            reconsider ab = a * b as Element of C by XBOOLE_0:def 3;
            per cases;
             suppose
A90:          c = o; then
              c in {o} by TARSKI:def 1; then
              reconsider c1 = c as Element of C by XBOOLE_0:def 3;
              per cases;
               suppose
A91:            (MULTR).(x,x) = x;
A92:            b * c = (multR(x,o)).(b1,c1) by Def8 .= multR(b1,c1) by Def7
                .= o by A90,A85,A91,Def6; then
                b * c in {o} by TARSKI:def 1; then
          reconsider bc = b * c as Element of C by XBOOLE_0:def 3;
                thus
                (a * b) * c = (multR(x,o)).(ab,c1) by Def8
                .= multR(ab,c1) by Def7 .= o by A89,A90,A91,Def6
                .= multR(a1,bc) by A86,A83,A92,Def6
                .= (multR(x,o)).(a,bc) by Def7
                .= a * (b * c) by Def8;
               end;
               suppose
A93:            (MULTR).(x,x) <> x;
A94:            b * c = (multR(x,o)).(b1,c1) by Def8 .= multR(b1,c1) by Def7
                .= x * x by A93,A90,A85,Def6; then
A95:            b * c <> o by A1; then
                not b * c in {o} by TARSKI:def 1; then
A96:            b * c in [#]F \ {x} by A2,XBOOLE_0:def 3; then
                reconsider bcR = b * c as Element of F;
                reconsider bc = b*c as Element of C by Def8;
A97:            (a * b) * c = (multR(x,o)).(ab,c1) by Def8
                .= multR(ab,c1) by Def7
                .= (aR * x) * x by A86,A89,A90,A93,Def6;
                now assume (MULTR).(a1,bc) = x; then
                 aR * bcR = aR * x by A86; then
                 bcR = x by A88,ALGSTR_0:def 36,ALGSTR_0:def 20; then
                 b * c in {x} by TARSKI:def 1;
                 hence contradiction by A96,XBOOLE_0:def 5;
                end; then
                (MULTR).(a1,bc) = multR(a1,bc) by A83,A95,Def6
                .= (multR(x,o)).(a,bc) by Def7 .= a * (b * c) by Def8;
                hence a * (b * c) = aR * (x * x) by A94
                .= (a * b) * c by A97,GROUP_1:def 3;
              end;
            end;
            suppose
A99:         c <> o; then
             not c in {o} by TARSKI:def 1; then
A100:        c in [#]F \ {x} by A2,XBOOLE_0:def 3;
             reconsider c1 = c as Element of C by Def8;
             reconsider cR = c as Element of F by A100;
             per cases;
              suppose
A101:          (MULTR).(x,c) = x;
A102:          b * c = (multR(x,o)).(b1,c1) by Def8 .= multR(b1,c1) by Def7
               .= o by A101,A99,A85,Def6; then
               b * c in {o} by TARSKI:def 1; then
               reconsider bc = b * c as Element of C by XBOOLE_0:def 3;
               thus
               (a * b) * c = (multR(x,o)).(ab,c1) by Def8
               .= multR(ab,c1) by Def7 .= o by A89,A99,A101,Def6
               .= multR(a1,bc) by A86,A83,A102,Def6
               .= (multR(x,o)).(a,bc) by Def7
               .= a * (b * c) by Def8;
              end;
              suppose
A103:          (MULTR).(x,c) <> x;
A104:          b * c = (multR(x,o)).(b1,c1) by Def8 .= multR(b1,c1) by Def7
               .= x * cR by A103,A99,A85,Def6; then
A105:          b * c <> o by A1; then
               not b * c in {o} by TARSKI:def 1; then
A106:          b * c in [#]F \ {x} by A2,XBOOLE_0:def 3;
               reconsider bc = b * c as Element of C by Def8;
               reconsider bcR = b * c as Element of F by A106;
A107:          now assume (MULTR).(a1,bc) = x; then
                aR * bcR = aR * x by A86; then
                bcR = x by A88,ALGSTR_0:def 36,ALGSTR_0:def 20; then
                b * c in {x} by TARSKI:def 1;
                hence contradiction by A106,XBOOLE_0:def 5;
               end;
               thus
               (a * b) * c = (multR(x,o)).(ab,c1) by Def8
               .= multR(ab,c1) by Def7
               .= (aR * x) * cR by A86,A89,A99,A103,Def6
               .= aR * (x * cR) by GROUP_1:def 3
               .= multR(a1,bc) by A107,A105,A83,A104,Def6
               .= (multR(x,o)).(a,bc) by Def7
               .= a * (b * c) by Def8;
              end;
             end;
            end;
            suppose
A109:        (MULTR).(a1,x) <> x;
A110:        a * b = (multR(x,o)).(a1,b1) by Def8 .= multR(a1,b1) by Def7
             .= aR * x by A85,A83,A109,Def6; then
A111:        a * b <> o by A1; then
             not a * b in {o} by TARSKI:def 1; then
A112:        a * b in [#]F \ {x} by A2,XBOOLE_0:def 3;
             reconsider ab = a * b as Element of C by Def8;
             reconsider abR = a * b as Element of F by A112;
             per cases;
              suppose
A113:          c = o; then
               c in {o} by TARSKI:def 1; then
               reconsider c1 = c as Element of C by XBOOLE_0:def 3;
               per cases;
                suppose
A114:            (MULTR).(x,x) = x;
A115:            b * c = (multR(x,o)).(b1,c1) by Def8 .= multR(b1,c1) by Def7
                 .= o by A114,A113,A85,Def6; then
                 b * c in {o} by TARSKI:def 1; then
                 reconsider bc = b * c as Element of C by XBOOLE_0:def 3;
A116:            now assume (MULTR).(ab,x) = x; then
                  x = (aR * x) * x by A110
                  .= aR * (x * x) by GROUP_1:def 3
                  .= aR * x by A114;
                  hence contradiction by A109;
                 end;
                 thus
                 (a * b) * c = (multR(x,o)).(ab,c1) by Def8
                 .= multR(ab,c1) by Def7
                 .= (aR * x) * x by A110,A1,A113,A116,Def6
                 .= aR * (x * x) by GROUP_1:def 3
                 .= multR(a1,bc) by A114,A109,A83,A115,Def6
                 .= (multR(x,o)).(a,bc) by Def7 .= a * (b * c) by Def8;
                end;
                suppose
A118:            (MULTR).(x,x) <> x;
A119:            b * c = (multR(x,o)).(b1,c1) by Def8 .= multR(b1,c1) by Def7
                 .= x * x by A118,A113,A85,Def6; then
A120:            b * c <> o by A1; then
                 not b * c in {o} by TARSKI:def 1; then
A121:            b * c in [#]F \ {x} by A2,XBOOLE_0:def 3;
                 reconsider bc = b * c as Element of C by Def8;
                 reconsider bcR = b * c as Element of F by A121;
A122:            (MULTR).(a,bc) = aR * (x * x) by A119
                 .= (aR * x) * x by GROUP_1:def 3 .= (MULTR).(ab,x) by A110;
                 per cases;
                  suppose
A123:              (MULTR).(ab,x) = x;
                   thus (a * b) * c = (multR(x,o)).(ab,c1) by Def8
                   .= multR(ab,c1) by Def7 .= o by A111,A113,A123,Def6
                   .= multR(a1,bc) by A123,A83,A122,A120,Def6
                   .= (multR(x,o)).(a,bc) by Def7
                   .= a * (b * c) by Def8;
                  end;
                  suppose
A124:              (MULTR).(ab,x) <> x;
                   thus (a * b) * c = (multR(x,o)).(ab,c1) by Def8
                   .= multR(ab,c1) by Def7
                   .= (MULTR).(ab,x) by A110,A1,A113,A124,Def6
                   .= multR(a1,bc) by A124,A83,A122,A120,Def6
                   .= (multR(x,o)).(a,bc) by Def7
                   .= a * (b * c) by Def8;
                  end;
                 end;
                end;
                suppose
A125:            c <> o; then
                 not c in {o} by TARSKI:def 1; then
A126:            c in [#]F \ {x} by A2,XBOOLE_0:def 3;
                 reconsider c1 = c as Element of C by Def8;
                 reconsider cR = c as Element of F by A126;
                 per cases;
                  suppose
A127:              (MULTR).(x,c) = x;
A128:              b * c = (multR(x,o)).(b1,c1) by Def8 .= multR(b1,c1) by Def7
                   .= o by A127,A125,A85,Def6; then
                   b * c in {o} by TARSKI:def 1; then
                   reconsider bc = b * c as Element of C by XBOOLE_0:def 3;
A129:              (MULTR).(ab,c) = (aR * x) * cR by A110
                   .= aR * (x * cR) by GROUP_1:def 3 .= (MULTR).(a,x) by A127;
                   per cases;
                    suppose
A130:                (MULTR).(ab,c1) = x;
                     thus (a * b) * c = (multR(x,o)).(ab,c1) by Def8
                     .= multR(ab,c1) by Def7 .= o by A111,A125,A130,Def6
                     .= multR(a1,bc) by A130,A83,A128,A129,Def6
                     .= (multR(x,o)).(a,bc) by Def7
                     .= a * (b * c) by Def8;
                    end;
                    suppose
A131:                (MULTR).(ab,c1) <> x;
                     thus (a * b) * c = (multR(x,o)).(ab,c1) by Def8
                     .= multR(ab,c1) by Def7 .= (aR * x) * cR
                      by A110,A111,A125,A131,Def6
                     .= aR * (x * cR) by GROUP_1:def 3
                     .= multR(a1,bc) by A127,A109,A83,A128,Def6
                     .= (multR(x,o)).(a,bc) by Def7
                     .= a * (b * c) by Def8;
                    end;
                   end;
                   suppose
A132:               (MULTR).(x,c) <> x;
A133:               b * c = (multR(x,o)).(b1,c1) by Def8
                    .= multR(b1,c1) by Def7
                    .= x * cR by A132,A125,A85,Def6; then
A134:               b * c <> o by A1; then
                    not b * c in {o} by TARSKI:def 1; then
A135:               b * c in [#]F \ {x} by A2,XBOOLE_0:def 3;
                    reconsider bc = b * c as Element of C by Def8;
                    reconsider bcR = b * c as Element of F by A135;
A136:               (MULTR).(a,bc) = aR * (x * cR) by A133
                    .= (aR * x) * cR by GROUP_1:def 3
                    .= (MULTR).(ab,c) by A110;
                    per cases;
                     suppose
A137:                 (MULTR).(ab,c1) = x;
                      thus (a * b) * c = (multR(x,o)).(ab,c1) by Def8
                      .= multR(ab,c1) by Def7 .= o by A111,A125,A137,Def6
                      .= multR(a1,bc) by A137,A83,A136,A134,Def6
                      .= (multR(x,o)).(a,bc) by Def7
                      .= a * (b * c) by Def8;
                     end;
                     suppose
A138:                 (MULTR).(ab,c1) <> x;
                      thus (a * b) * c = (multR(x,o)).(ab,c1) by Def8
                      .= multR(ab,c1) by Def7 .= (aR * x) * cR
                       by A110,A111,A125,A138,Def6
                      .= aR * (x * cR) by GROUP_1:def 3
                      .= multR(a1,bc) by A138,A83,A133,A136,A134,Def6
                      .= (multR(x,o)).(a,bc) by Def7 .= a * (b * c) by Def8;
                     end;
                    end;
                   end;
                  end;
                 end;
                 suppose
A139:             b <> o; then
                  not b in {o} by TARSKI:def 1; then
A140:             b in [#]F \ {x} by A2,XBOOLE_0:def 3;
                  reconsider b1 = b as Element of C by Def8;
                  reconsider bR = b as Element of F by A140;
A141:             now assume
A142:              x = x * x;
                   x <> 0.F by Def2; then
A143:              x is left_mult-cancelable by ALGSTR_0:def 36;
                   x * x = x * 1.F by A142;
                   hence contradiction by A143,ALGSTR_0:def 20,Def2;
                  end;
                  per cases;
                   suppose
A144:               (MULTR).(a,b) = x;
A145:               a * b = (multR(x,o)).(a1,b1) by Def8
                    .= multR(a1,b1) by Def7
                    .= o by A144,A139,A83,Def6; then
                    a * b in {o} by TARSKI:def 1; then
                    reconsider ab = a * b as Element of C by XBOOLE_0:def 3;
A146:               now assume bR = 0.F; then
                     aR * 0.F = x by A144;
                     hence contradiction by Def2;
                    end;
A148:               now assume bR * x = x; then
                     bR * x = aR * bR by A144
                     .= bR * aR by GROUP_1:def 12; then
                     x = aR by A146,ALGSTR_0:def 36,ALGSTR_0:def 20; then
                     a in {x} by TARSKI:def 1;
                     hence contradiction by A84,XBOOLE_0:def 5;
                    end;
                    per cases;
                     suppose
A150:                 c = o; then
                      c in {o} by TARSKI:def 1; then
                      reconsider c1 = c as Element of C by XBOOLE_0:def 3;
A151:                 b * c = (multR(x,o)).(b1,c1) by Def8
                      .= multR(b1,c1) by Def7
                      .= bR * x by A148,A150,A139,Def6; then
A152:                 b * c <> o by A1; then
                      not b * c in {o} by TARSKI:def 1; then
A153:                 b * c in [#]F \ {x} by A2,XBOOLE_0:def 3;
                      reconsider bc = b * c as Element of C by Def8;
                      reconsider bcR = b * c as Element of F by A153;
A154:                 now assume (MULTR).(a,bc) = x; then
                       x = aR * (bR * x) by A151
                       .= (aR * bR) * x by GROUP_1:def 3
                       .= x * x by A144;
                       hence contradiction by A141;
                      end;
                      thus
                      (a * b) * c = (multR(x,o)).(ab,c1) by Def8
                      .= multR(ab,c1) by Def7 .= (aR * bR) * x
                         by A144,A145,A150,A141,Def6
                      .= aR * (bR * x) by GROUP_1:def 3 .= multR(a1,bc)
                         by A83,A152,A151,A154,Def6
                      .= (multR(x,o)).(a,bc) by Def7 .= a * (b * c) by Def8;
                      end;
                      suppose
A156:                  c <> o; then
                       not c in {o} by TARSKI:def 1; then
A157:                  c in [#]F \ {x} by A2,XBOOLE_0:def 3;
                       reconsider c1 = c as Element of C by Def8;
                       reconsider cR = c as Element of F by A157;
                       per cases;
                        suppose
A158:                    (MULTR).(b,c) <> x;
A159:                    b * c = (multR(x,o)).(b1,c1) by Def8
                         .= multR(b1,c1) by Def7
                         .= bR * cR by A156,A139,A158,Def6; then
A160:                    b * c <> o by A1; then
                         not b * c in {o} by TARSKI:def 1; then
A161:                    b * c in [#]F\{x} by A2,XBOOLE_0:def 3;
                         reconsider bc = b*c as Element of C by Def8;
                         reconsider bcR = b*c as Element of F by A161;
                         per cases;
                          suppose
A162:                     (MULTR).(x,c) <> x;
A163:                     now assume (MULTR).(a,bc) = x; then
                           x = aR * (bR * cR) by A159
                           .= (aR * bR) * cR by GROUP_1:def 3
                           .= x * cR by A144;
                           hence contradiction by A162;
                          end;
                          thus
                          (a * b) * c = (multR(x,o)).(ab,c1) by Def8
                          .= multR(ab,c1) by Def7 .= (aR * bR) * cR
                             by A144,A145,A156,A162,Def6
                          .= aR * (bR * cR) by GROUP_1:def 3
                          .= multR(a1,bc) by A83,A163,A159,A160,Def6
                          .= (multR(x,o)).(a,bc) by Def7
                          .= a * (b * c) by Def8;
                         end;
                         suppose
A164:                     (MULTR).(x,c) = x;
A165:                     aR * (bR * cR) = (aR * bR) * cR by GROUP_1:def 3
                          .= x by A144,A164;
                          thus
                          (a * b) * c = (multR(x,o)).(ab,c1) by Def8
                          .= multR(ab,c1) by Def7 .= o by A164,A145,A156,Def6
                          .= multR(a1,bc) by A165,A83,A160,A159,Def6
                          .= (multR(x,o)).(a,bc) by Def7
                          .= a * (b * c) by Def8;
                         end;
                        end;
                        suppose
A166:                    (MULTR).(b,c) = x; then
                         bR * cR = aR * bR by A144
                         .= bR * aR by GROUP_1:def 12; then
A167:                    cR = aR by A146,ALGSTR_0:def 36,ALGSTR_0:def 20;
A168:                    b * c = (multR(x,o)).(b1,c1) by Def8
                         .= multR(b1,c1) by Def7
                         .= o by A156,A139,A166,Def6; then
                         b * c in {o} by TARSKI:def 1; then
                         reconsider bc = b*c as Element of C by XBOOLE_0:def 3;
A169:                    x * cR = aR * x by A167,GROUP_1:def 12
                         .= (MULTR).(a,x);
                         per cases;
                          suppose
A170:                      (MULTR).(x,c) <> x;
                           thus (a * b) * c = (multR(x,o)).(ab,c1) by Def8
                           .= multR(ab,c1) by Def7
                           .= x * cR by A145,A156,A170,Def6
                           .= multR(a1,bc) by A83,A169,A170,A168,Def6
                           .= (multR(x,o)).(a,bc) by Def7
                           .= a * (b * c) by Def8;
                          end;
                          suppose
A171:                      (MULTR).(x,c) = x;
                           thus (a * b) * c = (multR(x,o)).(ab,c1) by Def8
                           .= multR(ab,c1) by Def7 .= o by A145,A156,A171,Def6
                           .= multR(a1,bc) by A83,A169,A171,A168,Def6
                           .= (multR(x,o)).(a,bc) by Def7
                           .= a * (b * c) by Def8;
                          end;
                         end;
                        end;
                       end;
                       suppose
A172:                   (MULTR).(a,b) <> x;
A173:                   a * b = (multR(x,o)).(a1,b1) by Def8
                        .= multR(a1,b1) by Def7
                        .= aR * bR by A172,A139,A83,Def6; then
A174:                   a * b <> o by A1; then
                        not a * b in {o} by TARSKI:def 1; then
A175:                   a * b in [#]F \ {x} by A2,XBOOLE_0:def 3;
                        reconsider ab = a * b as Element of C by Def8;
                        reconsider abR = a * b as Element of F by A175;
                        per cases;
                         suppose
A176:                     c = o; then
                          c in {o} by TARSKI:def 1; then
                          reconsider c1 = c as Element of C by XBOOLE_0:def 3;
                          per cases;
                           suppose
A177:                       (MULTR).(b,x) <> x;
A178:                       b * c = (multR(x,o)).(b1,c1) by Def8
                            .= multR(b1,c1) by Def7
                            .= bR * x by A176,A139,A177,Def6; then
A179:                       b * c <> o by A1; then
                            not b * c in {o} by TARSKI:def 1; then
A180:                       b * c in [#]F \ {x} by A2,XBOOLE_0:def 3;
                            reconsider bc = b*c as Element of C by Def8;
                            reconsider bcR = b*c as Element of F by A180;
A181:                       (MULTR).(ab,x) = (aR * bR) * x by A173
                            .= aR * (bR * x) by GROUP_1:def 3
                            .= (MULTR).(a,bc) by A178;
                            per cases;
                             suppose
A182:                         (MULTR).(ab,x) <> x;
                              thus
                              (a * b) * c = (multR(x,o)).(ab,c1) by Def8
                              .= multR(ab,c1) by Def7
                              .= (aR * bR) * x by A173,A1,A176,A182,Def6
                              .= aR * (bR * x) by GROUP_1:def 3
                              .= multR(a1,bc) by A83,A182,A181,A178,A179,Def6
                              .= (multR(x,o)).(a,bc) by Def7
                              .= a * (b * c) by Def8;
                             end;
                             suppose
A183:                         (MULTR).(ab,x) = x;
                              thus
                              (a * b) * c = (multR(x,o)).(ab,c1) by Def8
                              .= multR(ab,c1) by Def7
                              .= o by A174,A176,A183,Def6
                              .= multR(a1,bc) by A83,A183,A181,A179,Def6
                              .= (multR(x,o)).(a,bc) by Def7
                              .= a * (b * c) by Def8;
                             end;
                            end;
                            suppose
A184:                        (MULTR).(b,x) = x;
A185:                        b * c = (multR(x,o)).(b1,c1) by Def8
                             .= multR(b1,c1) by Def7
                             .= o by A176,A139,A184,Def6; then
                             b * c in {o} by TARSKI:def 1; then
                         reconsider bc = b*c as Element of C by XBOOLE_0:def 3;
A186:                        (aR * bR) * x = aR * (bR * x) by GROUP_1:def 3
                             .= (MULTR).(a,x) by A184;
                             per cases;
                              suppose
A187:                          (MULTR).(ab,x) <> x;
                               thus
                               (a * b) * c = (multR(x,o)).(ab,c1) by Def8
                               .= multR(ab,c1) by Def7
                               .= (aR * bR) * x by A173,A1,A176,A187,Def6
                               .= multR(a1,bc) by A173,A83,A187,A185,A186,Def6
                               .= (multR(x,o)).(a,bc) by Def7
                               .= a * (b * c) by Def8;
                              end;
                              suppose
A188:                          (MULTR).(ab,x) = x;
                               thus
                               (a * b) * c = (multR(x,o)).(ab,c1) by Def8
                               .= multR(ab,c1) by Def7
                               .= o by A174,A176,A188,Def6
                               .= multR(a1,bc) by A173,A83,A188,A185,A186,Def6
                               .= (multR(x,o)).(a,bc) by Def7
                               .= a * (b * c) by Def8;
                              end;
                             end;
                            end;
                            suppose
A189:                        c <> o; then
                             not c in {o} by TARSKI:def 1; then
A190:                        c in [#]F \ {x} by A2,XBOOLE_0:def 3;
                             reconsider c1 = c as Element of C by Def8;
                             reconsider cR = c as Element of F by A190;
                             per cases;
                              suppose
A191:                          (MULTR).(b,c) <> x;
A192:                          b * c = (multR(x,o)).(b1,c1) by Def8
                               .= multR(b1,c1) by Def7
                               .= bR * cR by A189,A139,A191,Def6; then
A193:                          b * c <> o by A1; then
                               not b * c in {o} by TARSKI:def 1; then
A194:                          b * c in [#]F\{x} by A2,XBOOLE_0:def 3;
                               reconsider bc = b*c as Element of C by Def8;
                               reconsider bcR = b*c as Element of F by A194;
A195:                          (MULTR).(ab,c) = (aR * bR) * cR  by A173
                               .= aR * (bR * cR) by GROUP_1:def 3
                               .= (MULTR).(a,bc) by A192;
                               per cases;
                                suppose
A196:                            (MULTR).(ab,c) <> x;
                                 thus
                                 (a * b) * c = (multR(x,o)).(ab,c1) by Def8
                                 .= multR(ab,c1) by Def7
                                 .= (aR * bR) * cR by A173,A174,A189,A196,Def6
                                 .= aR * (bR * cR) by GROUP_1:def 3
                                 .= multR(a1,bc)
                                    by A193,A83,A196,A192,A195,Def6
                                 .= (multR(x,o)).(a,bc) by Def7
                                 .= a * (b * c) by Def8;
                                end;
                                suppose
A197:                            (MULTR).(ab,c) = x;
                                 thus
                                 (a * b) * c = (multR(x,o)).(ab,c1) by Def8
                                 .= multR(ab,c1) by Def7
                                 .= o by A174,A189,A197,Def6
                                 .= multR(a1,bc) by A193,A83,A197,A195,Def6
                                 .= (multR(x,o)).(a,bc) by Def7
                                 .= a * (b * c) by Def8;
                                end;
                               end;
                               suppose
A198:                           (MULTR).(b,c) = x;
A199:                           b * c = (multR(x,o)).(b1,c1) by Def8
                                .= multR(b1,c1) by Def7
                                .= o by A189,A139,A198,Def6; then
                                b * c in {o} by TARSKI:def 1; then
                     reconsider bc = b*c as Element of C by XBOOLE_0:def 3;
A200:                           (aR * bR) * cR
                                = aR * (bR * cR) by GROUP_1:def 3
                                .= (MULTR).(a,x) by A198;
                                per cases;
                                 suppose
A201:                             (MULTR).(ab,c) <> x;
                                  thus
                                  (a * b) * c = (multR(x,o)).(ab,c1) by Def8
                                  .= multR(ab,c1) by Def7
                                  .= (aR * bR) * cR by A173,A174,A189,A201,Def6
                                  .= multR(a1,bc)
                                     by A173,A83,A201,A199,A200,Def6
                                  .= (multR(x,o)).(a,bc) by Def7
                                  .= a * (b * c) by Def8;
                                 end;
                                 suppose
A202:                             (MULTR).(ab,c) = x;
                                  thus
                                  (a * b) * c = (multR(x,o)).(ab,c1) by Def8
                                  .= multR(ab,c1) by Def7
                                  .= o by A174,A189,A202,Def6
                                  .= multR(a1,bc)
                                     by A173,A83,A202,A199,A200,Def6
                                  .= (multR(x,o)).(a,bc) by Def7
                                  .= a * (b * c) by Def8;
                                 end;
                                end;
                               end;
                              end;
                             end;
                            end;
                           end;
                           hence ExField(x,o) is associative;
                          end;
