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

theorem Th8:
   for x being non trivial Element of F, o being object st not o in [#]F
   holds ExField(x,o) is add-associative
   proof
    let x be non trivial Element of F, o be object;
    assume a1: not o in [#]F; then
A1: a <> o;
    set C = carr(x,o), E = ExField(x,o);
    set ADDR = the addF 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:        (ADDR).(x,x) <> x;
A6:        a + b = (addR(x,o)).(a1,b1) by Def8
           .= addR(a1,b1) by Def5 .= x + x by A3,A4,A5,Def4;
           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 = (addR(x,o)).(a+b,c1) by Def8
             .= addR(xx,c1) by A6,Def5;
A10:         b + c = (addR(x,o)).(b1,c1) by Def8
             .= addR(b1,c1) by Def5 .= x + x by A4,A5,A8,Def4;
             per cases;
              suppose
A11:           (ADDR).(xx,x) <> x;
A12:           (ADDR).(x,xx) = x + (x + x) .= (x + x) + x;
               thus (a + b) + c = (x + x) + x by A1,A8,A9,A11,Def4
               .= addR(a1,xx) by A1,A3,A11,A12,Def4
               .= (addR(x,o)).(a1,xx) by Def5
               .= a + (b + c) by A10,Def8;
              end;
              suppose
A13:           (ADDR).(xx,x) = x;
A14:           (ADDR).(x,xx) = x + (x + x) .= (x + x) + x;
               thus (a + b) + c = o by A7,A8,A9,A13,Def4
               .= addR(a1,xx) by A3,A7,A13,A14,Def4
               .= (addR(x,o)).(a1,xx) by Def5
               .= a + (b + c) by A10,Def8;
              end;
             end;
             suppose
A15:          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;
A16:          (a + b) + c = (addR(x,o)).(a+b,c1) by Def8
              .= addR(xx,c1) by A6,Def5;
A17:          b + c = (addR(x,o)).(b1,c1) by Def8
              .= addR(b1,c1) by Def5; then
              reconsider bc = b + c as Element of C;
              per cases;
               suppose
A18:            (ADDR).(x,c1) <> x; then
A19:            b + c = x + cR by A4,A15,A17,Def4; then
A20:            b + c <> o by a1;
                per cases;
                 suppose (ADDR).(xx,c1) <> x; then
A21:              (a + b) + c = (x + x) + cR by A7,A15,A16,Def4
                  .= x + (x + cR) by RLVECT_1:def 3
                  .= (ADDR).(x,b+c) by A4,A15,A17,A18,Def4;
                  per cases;
                   suppose (ADDR).(x,bc) <> x;
                    hence (a + b) + c = addR(a1,bc) by A1,A3,A19,A21,Def4
                    .= (addR(x,o)).(a1,b+c) by Def5 .= a + (b + c) by Def8;
                   end;
                   suppose
A22:                (ADDR).(x,bc) = x;
A23:                (ADDR).(x,bc) = x + (x + cR) by A4,A15,A17,A18,Def4
                    .=(x + x) + cR by RLVECT_1:def 3.=(ADDR).(xx,c1);
                    thus
                    (a + b) + c = o by A7,A15,A16,A22,A23,Def4
                    .= addR(a1,bc) by A3,A20,A22,Def4
                    .= (addR(x,o)).(a1,b+c) by Def5 .= a + (b + c) by Def8;
                   end;
                  end;
                  suppose
A24:               (ADDR).(xx,c1) = x; then
A25:               (a + b) + c = o by A7,A15,A16,Def4;
                   per cases;
                    suppose (ADDR).(x,bc) <> x;
A26:                 (ADDR).(x,bc) = x + (x + cR) by A4,A15,A17,A18,Def4
                     .= (x + x) + cR by RLVECT_1:def 3
                     .= (ADDR).(xx,c1);
                     thus
                     (a + b) + c = addR(a1,bc) by A3,A20,A26,A24,A25,Def4
                     .= (addR(x,o)).(a1,b+c) by Def5 .= a + (b + c) by Def8;
                    end;
                    suppose
A27:                 (ADDR).(x,bc) = x;
                     (ADDR).(x,bc) = x + (x + cR) by A4,A15,A17,A18,Def4
                     .= (x + x) + cR by RLVECT_1:def 3
                     .= (ADDR).(xx,c1);
                     hence (a + b) + c = o by A7,A15,A16,A27,Def4
                     .= addR(a1,bc) by A3,A20,A27,Def4
                     .= (addR(x,o)).(a1,b+c) by Def5 .= a + (b + c) by Def8;
                    end;
                   end;
                  end;
                  suppose
A29:               (ADDR).(x,c1) = x; then
                   x + cR = x; then
A30:               c1 = 0.F by RLVECT_1:9;
A31:               b + c = o by A4,A29,A15,A17,Def4;
                   per cases;
                    suppose (ADDR).(xx,c1) <> x;
                     hence (a + b) + c =(x + x)+ cR by A7,A15,A16,Def4
                     .= addR(a1,bc) by A3,A5,A30,A31,Def4
                     .= (addR(x,o)).(a1,b+c) by Def5
                     .= a + (b + c) by Def8;
                    end;
                    suppose (ADDR).(xx,c1) = x; then
                     x = (x + x) + cR .= x + x by A30;
                     hence (a + b) + c = a + (b + c) by A5;
                    end;
                   end;
                  end;
                 end;
                 suppose
A33:              (ADDR).(x,x) = x;
A34:              a + b = (addR(x,o)).(a1,b1) by Def8
                  .= addR(a1,b1) by Def5.= o by A3,A4,A33,Def4;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 c = o;
                    hence (a + b) + c = a + (b + c) by A3;
                   end;
                   suppose
A36:                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
A37:                  (ADDR).(x,c1) = x;
A38:                  b + c = (addR(x,o)).(b1,c1) by Def8
                      .= addR(b1,c1) by Def5 .= o by A4,A36,A37,Def4;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 = (addR(x,o)).(ab,c1) by Def8
                      .= addR(ab,c1) by Def5 .= o by A36,A34,A37,Def4
                      .= addR(a1,bc) by A3,A33,A38,Def4
                      .= (addR(x,o)).(a1,b+c) by Def5 .= a + (b + c) by Def8;
                     end;
                     suppose
A39:                  (ADDR).(x,c1) <> x;
A40:                  b + c = (addR(x,o)).(b1,c1) by Def8
                      .= addR(b1,c1) by Def5 .= x + cR by A4,A36,A39,Def4;
                      reconsider bc = b+c as Element of C by Def8;
A41:                  (a + b) + c = (addR(x,o)).(ab,c1) by Def8
                      .= addR(ab,c1) by Def5 .= (ADDR).(x,c1)
                      by A34,A36,A39,Def4;
                      (a + b) + c = (addR(x,o)).(ab,c1) by Def8
                      .= addR(ab,c1) by Def5
                      .= (x + x) + cR by A33,A34,A36,A39,Def4
                      .= x + (x + cR) by RLVECT_1:def 3
                      .= (ADDR).(x,bc) by A40;
                      hence (a + b) + c = addR(a1,bc) by A1,A3,A39,A40,A41,Def4
                      .= (addR(x,o)).(a1,b+c) by Def5 .= a + (b + c) by Def8;
                     end;
                    end;
                   end;
                  end;
                  suppose
A42:               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;
A43:               (ADDR).(x,b) = x + bR .= bR + x
                   .= (ADDR).(b,x);
                   per cases;
                    suppose
A44:                 (ADDR).(x,b) <> x;
A45:                 a + b = (addR(x,o)).(a1,b1) by Def8
                     .= addR(a1,b1) by Def5
                     .= x + bR by A3,A42,A44,Def4; then
A46:                 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
A47:                   c = o; then
                       c in {o} by TARSKI:def 1; then
                       reconsider c1 = c as Element of C by XBOOLE_0:def 3;
A48:                   b + c = (addR(x,o)).(b1,c1) by Def8
                       .= addR(b1,c1) by Def5
                       .= bR + x by A42,A43,A44,A47,Def4;
A49:                   b + c <> o by A1,A48; 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;
A50:                   (ADDR).(ab,x) = (x + bR) + x by A45 .= x + (bR + x);
                       per cases;
                        suppose
A51:                     (ADDR).(ab,x) <> x;
                         thus (a + b) + c = (addR(x,o)).(ab,c1) by Def8
                         .= addR(ab,c1) by Def5
                         .= (ADDR).(x,bc) by A1,A3,A47,A48,A50,A51,Def4
                         .= addR(a1,bc) by A1,A3,A48,A50,A51,Def4
                         .= (addR(x,o)).(a,bc) by Def5
                         .= a + (b + c) by Def8;
                        end;
                        suppose
A52:                     (ADDR).(ab,x) = x;
                         thus (a + b) + c = (addR(x,o)).(ab,c1) by Def8
                         .= addR(ab,c1) by Def5 .= o by A46,A47,A52,Def4
                         .= addR(a1,bc) by A3,A48,A49,A50,A52,Def4
                         .= (addR(x,o)).(a,bc) by Def5
                         .= a + (b + c) by Def8;
                        end;
                       end;
                       suppose
A53:                    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;
A54:                    (ADDR).(ab,c) = (x + bR) + cR by A45
                        .= x + (bR + cR) by RLVECT_1:def 3;
                        per cases;
                         suppose
A55:                      (ADDR).(b,c) <> x;
A56:                      b + c = (addR(x,o)).(b1,c1) by Def8
                          .= addR(b1,c1) by Def5
                          .= bR + cR by A42,A53,A55,Def4;
A57:                      b + c <> o by A1,A56; 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
A58:                        (ADDR).(ab,c1) <> x;
                            thus (a + b) + c =(addR(x,o)).(ab,c1) by Def8
                            .= addR(ab,c1) by Def5
                            .= (x + bR) + cR by A45,A58,A53,A46,Def4
                            .= x + (bR + cR) by RLVECT_1:def 3
                            .= addR(a1,bc) by A1,A3,A54,A58,A56,Def4
                            .= (addR(x,o)).(a,bc) by Def5
                            .= a + (b + c) by Def8;
                           end;
                           suppose
A59:                        (ADDR).(ab,c1) = x;
                            thus (a + b) + c = (addR(x,o)).(ab,c1) by Def8
                            .= addR(ab,c1) by Def5 .= o by A59,A53,A46,Def4
                            .= addR(a1,bc) by A3,A54,A59,A56,A57,Def4
                            .= (addR(x,o)).(a,bc) by Def5
                            .= a + (b + c) by Def8;
                           end;
                          end;
                          suppose
A60:                       (ADDR).(b,c) = x;
A61:                       b + c = (addR(x,o)).(b1,c1) by Def8
                           .= addR(b1,c1) by Def5
                           .= o by A60,A53,A42,Def4; 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
A62:                         (ADDR).(ab,c1) <> x;
                             thus (a + b) + c = (addR(x,o)).(ab,c1) by Def8
                             .= addR(ab,c1) by Def5
                             .= (x + bR) + cR by A45,A46,A53,A62,Def4
                             .= x + (bR + cR) by RLVECT_1:def 3
                             .= addR(a1,bc) by A60,A3,A54,A62,A61,Def4
                             .= (addR(x,o)).(a,bc) by Def5
                             .= a + (b + c) by Def8;
                            end;
                            suppose
A63:                         (ADDR).(ab,c1) = x;
                             thus (a + b)+ c = (addR(x,o)).(ab,c1) by Def8
                             .= addR(ab,c1) by Def5 .= o by A63,A53,A46,Def4
                             .= addR(a1,bc) by A60,A3,A54,A63,A61,Def4
                             .= (addR(x,o)).(a,bc) by Def5
                             .= a + (b + c) by Def8;
                            end;
                           end;
                          end;
                         end;
                         suppose
A64:                      (ADDR).(x,b) = x;
A65:                      a + b=(addR(x,o)).(a1,b1) by Def8
                          .= addR(a1,b1) by Def5
                          .= o by A64,A42,A3,Def4; 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 c = o;
                            hence (a + b) + c = a + (b + c) by A3;
                           end;
                           suppose
A68:                        c <> o; then
                            not c in {o} by TARSKI:def 1; then
A69:                        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;
A70:                        now assume (ADDR).(b,c) = x; then
                             bR + cR = x + bR by A64 .= bR + x; then
                             x = cR by ALGSTR_0:def 4;then
                             c in {x} by TARSKI:def 1;
                             hence contradiction by A69,XBOOLE_0:def 5;
                            end;
A72:                        b + c = (addR(x,o)).(b1,c1) by Def8
                            .= addR(b1,c1) by Def5
                            .= bR + cR by A70,A68,A42,Def4;
A73:                        b + c <> o by A72,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;
A74:                        x + (bR + cR) = (x + bR) + cR by RLVECT_1:def 3
                            .= x + cR by A64;
                            per cases;
                             suppose
A75:                          (ADDR).(x,c1) <> x;
                              thus (a + b)+ c = (addR(x,o)).(ab,c1) by Def8
                              .= addR(ab,c1) by Def5
                              .= (ADDR).(x,c1) by A65,A75,A68,Def4
                              .= addR(a1,bc) by A74,A3,A75,A72,A1,Def4
                              .= (addR(x,o)).(a,bc) by Def5
                              .= a + (b + c) by Def8;
                             end;
                             suppose
A76:                          (ADDR).(x,c1) = x;
                              thus (a + b) + c = (addR(x,o)).(ab,c1) by Def8
                              .= addR(ab,c1) by Def5
                              .= o by A76,A68,A65,Def4
                              .= addR(a1,bc) by A73,A76,A74,A3,A72,Def4
                              .= (addR(x,o)).(a,bc) by Def5
                              .= a + (b + c) by Def8;
                             end;
                            end;
                           end;
                          end;
                         end;
     suppose
A77:  a <> o;
      not a in {o} by A77,TARSKI:def 1; then
A78:  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
A79:    b = o;
        b in {o} by A79,TARSKI:def 1; then
       reconsider b1 = b as Element of C by XBOOLE_0:def 3;
        per cases;
         suppose
A80:      (ADDR).(a1,x) = x;
A81:      a + b = (addR(x,o)).(a1,b1) by Def8
          .= addR(a1,b1) by Def5
          .= o by A80,A79,A77,Def4; 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
A81a:       c = o;
            c in {o} by A81a,TARSKI:def 1; then
            reconsider c1 = c as Element of C by XBOOLE_0:def 3;
            per cases;
             suppose
A82:          (ADDR).(x,x) = x;
A83:          b + c =(addR(x,o)).(b1,c1) by Def8
              .= addR(b1,c1) by Def5
              .= o by A81a,A79,A82,Def4; 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 = (addR(x,o)).(ab,c1) by Def8
              .= addR(ab,c1) by Def5
              .= o by A81,A81a,A82,Def4
              .= addR(a1,bc) by A80,A77,A83,Def4
              .= (addR(x,o)).(a,bc) by Def5
              .= a + (b + c) by Def8;
             end;
              suppose
A84:           (ADDR).(x,x) <> x;
A85:           b + c = (addR(x,o)).(b1,c1) by Def8
               .= addR(b1,c1) by Def5
               .= x + x by A84,A79,A81a,Def4; then
A86:           b + c <> o by A1; then
               not b + c in {o} by TARSKI:def 1; then
A87:           b + c in [#]F\{x} by A2,XBOOLE_0:def 3;
               reconsider bcR = b + c as Element of F by A87;
               reconsider bc = b+c as Element of C by Def8;
A88:           (a + b) + c = (addR(x,o)).(ab,c1) by Def8
               .= addR(ab,c1) by Def5
               .= (aR + x) + x by A80,A81,A81a,A84,Def4;
               now assume (ADDR).(a1,bc) = x; then
                aR + bcR = aR + x by A80; then
                bcR = x by ALGSTR_0:def 4; then
                b + c in {x} by TARSKI:def 1;
                hence contradiction by A87,XBOOLE_0:def 5;
               end; then
               (ADDR).(a1,bc) = addR(a1,bc) by A77,A86,Def4
               .= (addR(x,o)).(a,bc) by Def5
               .= a + (b + c) by Def8;
               hence a + (b + c) = aR + (x + x) by A85
              .= (a + b) + c by A88,RLVECT_1:def 3;
             end;
            end;
            suppose
A90:         c <> o;
             not c in {o} by A90,TARSKI:def 1; then
A91:         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 A91;
             per cases;
              suppose
A92:           (ADDR).(x,c) = x;
A93:           b + c = (addR(x,o)).(b1,c1) by Def8
               .= addR(b1,c1) by Def5
               .= o by A92,A90,A79,Def4;
               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 = (addR(x,o)).(ab,c1) by Def8
               .=addR(ab,c1) by Def5
               .= o by A81,A90,A92,Def4
               .= addR(a1,bc) by A80,A77,A93,Def4
               .= (addR(x,o)).(a,bc) by Def5
               .= a + (b + c) by Def8;
              end;
              suppose
A94:           (ADDR).(x,c) <> x;
A95:           b + c = (addR(x,o)).(b1,c1) by Def8
               .= addR(b1,c1) by Def5
               .= x + cR by A94,A90,A79,Def4; then
A96:           b + c <> o by A1; then
               not b + c in {o} by TARSKI:def 1; then
A97:           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 A97;
A98:           now assume (ADDR).(a1,bc) = x; then
                aR + bcR = aR + x by A80; then
                bcR = x by ALGSTR_0:def 4; then
                b + c in {x} by TARSKI:def 1;
               hence contradiction by A97,XBOOLE_0:def 5;
              end;
              thus
              (a + b) + c = (addR(x,o)).(ab,c1) by Def8
              .= addR(ab,c1) by Def5
              .= (aR + x) + cR by A80,A81,A90,A94,Def4
              .= aR + (x + cR) by RLVECT_1:def 3
              .= addR(a1,bc) by A98,A96,A77,A95,Def4
              .= (addR(x,o)).(a,bc) by Def5
              .= a + (b + c) by Def8;
             end;
            end;
           end;
           suppose
A100:       (ADDR).(a1,x) <> x;
A101:       a + b = (addR(x,o)).(a1,b1) by Def8
            .= addR(a1,b1) by Def5
            .= aR + x by A79,A77,A100,Def4; then
A102:       a + b <> o by A1; then
            not a + b in {o} by TARSKI:def 1; then
A103:       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 A103;
            per cases;
             suppose
A104:         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
A105:           (ADDR).(x,x) = x;
A106:           b + c = (addR(x,o)).(b1,c1) by Def8
                .= addR(b1,c1) by Def5
                .= o by A105,A104,A79,Def4; then
                b + c in {o} by TARSKI:def 1; then
                reconsider bc = b + c as Element of C by XBOOLE_0:def 3;
A107:           now assume (ADDR).(ab,x) = x; then
                 x = (aR + x) + x by A101
                 .= aR + (x + x) by RLVECT_1:def 3
                 .= aR + x by A105;
                hence contradiction by A100;
               end;
               thus (a+b)+c = (addR(x,o)).(ab,c1) by Def8
                .= addR(ab,c1) by Def5
                .= (aR + x) + x by A101,A1,A104,A107,Def4
                .= aR + (x + x) by RLVECT_1:def 3
                .= addR(a1,bc) by A105,A100,A77,A106,Def4
                .= (addR(x,o)).(a,bc) by Def5
                .= a + (b + c) by Def8;
               end;
               suppose
A109:           (ADDR).(x,x) <> x;
A110:           b + c = (addR(x,o)).(b1,c1) by Def8
                .= addR(b1,c1) by Def5
                .= x + x by A109,A104,A79,Def4; then
A111:           b + c <> o by A1; then
                not b + c in {o} by TARSKI:def 1; then
A112:           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 A112;
A113:          (ADDR).(a,bc)=aR + (x + x) by A110
               .= (aR + x) + x by RLVECT_1:def 3
               .= (ADDR).(ab,x) by A101;
               per cases;
                suppose
A114:            (ADDR).(ab,x) = x;
                 thus (a + b) + c = (addR(x,o)).(ab,c1) by Def8
                 .= addR(ab,c1) by Def5
                 .= o by A102,A104,A114,Def4
                 .= addR(a1,bc) by A114,A77,A113,A111,Def4
                 .= (addR(x,o)).(a,bc) by Def5
                 .= a + (b + c) by Def8;
                end;
                suppose
A115:            (ADDR).(ab,x) <> x;
                 thus (a + b) + c = (addR(x,o)).(ab,c1) by Def8
                 .= addR(ab,c1) by Def5
                 .= (ADDR).(ab,x) by A101,A1,A104,A115,Def4
                 .= addR(a1,bc) by A115,A77,A113,A111,Def4
                 .= (addR(x,o)).(a,bc) by Def5
                 .= a + (b + c) by Def8;
                end;
               end;
              end;
              suppose
A116:          c <> o; then
               not c in {o} by TARSKI:def 1; then
A117:          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 A117;
               per cases;
                suppose
A118:            (ADDR).(x,c) = x;
A119:            b + c = (addR(x,o)).(b1,c1) by Def8
                 .= addR(b1,c1) by Def5
                 .= o by A118,A116,A79,Def4; then
                 b + c in {o} by TARSKI:def 1; then
                 reconsider bc = b + c as Element of C by XBOOLE_0:def 3;
A120:            (ADDR).(ab,c) = (aR + x) + cR by A101
                 .= aR + (x + cR) by RLVECT_1:def 3
                 .= (ADDR).(a,x) by A118;
                 per cases;
                  suppose
A121:              (ADDR).(ab,c1) = x;
                   thus (a + b) + c = (addR(x,o)).(ab,c1) by Def8
                   .= addR(ab,c1) by Def5
                   .= o by A102,A116,A121,Def4
                   .= addR(a1,bc) by A121,A77,A119,A120,Def4
                   .= (addR(x,o)).(a,bc) by Def5
                   .= a + (b + c) by Def8;
                  end;
                  suppose
A122:              (ADDR).(ab,c1) <> x;
                   thus (a + b) + c = (addR(x,o)).(ab,c1) by Def8
                   .= addR(ab,c1) by Def5
                   .= (aR + x) + cR by A101,A102,A116,A122,Def4
                   .= aR + (x + cR) by RLVECT_1:def 3
                   .= addR(a1,bc) by A119,A100,A77,A118,Def4
                   .= (addR(x,o)).(a,bc) by Def5
                   .= a + (b + c) by Def8;
                  end;
                 end;
                 suppose
A123:             (ADDR).(x,c) <> x;
A124:             b + c = (addR(x,o)).(b1,c1) by Def8
                  .= addR(b1,c1) by Def5
                  .= x + cR by A123,A116,A79,Def4; then
A125:             b + c <> o by A1; then
                  not b + c in {o} by TARSKI:def 1; then
A126:             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 A126;
A127:             (ADDR).(a,bc) = aR + (x + cR) by A124
                  .= (aR + x) + cR by RLVECT_1:def 3
                  .= (ADDR).(ab,c) by A101;
                  per cases;
                   suppose
A128:               (ADDR).(ab,c1) = x;
                    thus (a + b) + c = (addR(x,o)).(ab,c1) by Def8
                    .= addR(ab,c1) by Def5
                    .= o by A102,A116,A128,Def4
                    .= addR(a1,bc) by A128,A77,A125,A127,Def4
                    .= (addR(x,o)).(a,bc) by Def5
                    .= a + (b + c) by Def8;
                   end;
                   suppose
A129:               (the addF of F).(ab,c1) <> x;
                    thus (a + b) + c = (addR(x,o)).(ab,c1) by Def8
                    .= addR(ab,c1) by Def5
                    .= (aR + x) + cR by A101,A102,A116,A129,Def4
                    .= aR + (x + cR) by RLVECT_1:def 3
                    .= addR(a1,bc) by A129,A77,A124,A127,A125,Def4
                    .= (addR(x,o)).(a,bc) by Def5
                    .= a + (b + c) by Def8;
                   end;
                  end;
                 end;
                end;
               end;
               suppose
A130:           b <> o; then
                not b in {o} by TARSKI:def 1; then
A131:           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 A131;
A132:           now assume x = x + x; then
                 x = 0.F by RLVECT_1:9;
                hence contradiction by Def2;
               end;
               per cases;
                suppose
A134:            (ADDR).(a,b) = x;
A135:            a + b = (addR(x,o)).(a1,b1) by Def8
                 .= addR(a1,b1) by Def5
                 .= o by A134,A130,A77,Def4; then
                 a + b in {o} by TARSKI:def 1; then
                 reconsider ab = a + b as Element of C by XBOOLE_0:def 3;
A136:            now assume bR + x = x; then
A138:             bR + x = aR + bR by A134
                  .= bR + aR;
                  x = aR by A138,ALGSTR_0:def 4; then
                  a in {x} by TARSKI:def 1;
                  hence contradiction by A78,XBOOLE_0:def 5;
                 end;
                 per cases;
                  suppose
A139:              c = o; then
                   c in {o} by TARSKI:def 1; then
                   reconsider c1 = c as Element of C by XBOOLE_0:def 3;
A140:              b + c = (addR(x,o)).(b1,c1) by Def8
                   .= addR(b1,c1) by Def5
                   .= bR + x by A136,A139,A130,Def4; then
A141:              b + c <> o by A1; then
                   not b + c in {o} by TARSKI:def 1; then
A142:              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 A142;
A143:              now assume (ADDR).(a,bc) = x; then
                    x = aR + (bR + x) by A140
                    .= (aR + bR) + x by RLVECT_1:def 3
                    .= x + x by A134;
                    hence contradiction by A132;
                   end;
                   thus
                   (a + b) + c = (addR(x,o)).(ab,c1) by Def8
                   .= addR(ab,c1) by Def5
                   .= (aR + bR) + x by A134,A135,A139,A132,Def4
                   .= aR + (bR + x) by RLVECT_1:def 3
                   .= addR(a1,bc) by A77,A141,A140,A143,Def4
                   .= (addR(x,o)).(a,bc) by Def5
                   .= a + (b + c) by Def8;
                  end;
                  suppose
A145:              c <> o; then
                   not c in {o} by TARSKI:def 1; then
A146:              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 A146;
                   per cases;
                    suppose
A147:                (ADDR).(b,c) <> x;
A148:                b + c = (addR(x,o)).(b1,c1) by Def8
                     .= addR(b1,c1) by Def5
                     .= bR + cR by A145,A130,A147,Def4; then
A149:                b + c <> o by A1;
                     not b + c in {o} by A149,TARSKI:def 1; then
A150:                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 A150;
                     per cases;
                      suppose
A151:                  (ADDR).(x,c) <> x;
A152:                  now assume (ADDR).(a,bc) = x; then
                       x = aR + (bR + cR) by A148
                       .=(aR + bR) + cR by RLVECT_1:def 3
                       .= x + cR by A134;
                       hence contradiction by A151;
                      end;
                      thus
                      (a + b) + c = (addR(x,o)).(ab,c1) by Def8
                      .= addR(ab,c1) by Def5
                      .= (aR + bR) + cR by A134,A135,A145,A151,Def4
                      .= aR + (bR + cR) by RLVECT_1:def 3
                      .= addR(a1,bc) by A77,A148,A149,A152,Def4
                      .= (addR(x,o)).(a,bc) by Def5
                      .= a + (b + c) by Def8;
                     end;
                     suppose
A154:                 (ADDR).(x,c) = x;
A155:                 aR + (bR + cR) = (aR + bR) + cR by RLVECT_1:def 3
                      .= x by A134,A154;
                      thus (a + b) + c = (addR(x,o)).(ab,c1) by Def8
                      .= addR(ab,c1) by Def5
                      .= o by A154,A135,A145,Def4
                      .= addR(a1,bc) by A155,A77,A149,A148,Def4
                      .= (addR(x,o)).(a,bc) by Def5
                      .= a + (b + c) by Def8;
                     end;
                    end;
                    suppose
A156:                (ADDR).(b,c) = x; then
A157:                 bR + cR = aR + bR by A134 .= bR + aR;
A158:                 b + c = (addR(x,o)).(b1,c1) by Def8
                      .= addR(b1,c1) by Def5
                      .= o by A130,A145,A156,Def4; then
                      b + c in {o} by TARSKI:def 1; then
                      reconsider bc = b+c as Element of C by XBOOLE_0:def 3;
A159:                 x + cR = aR + x by A157,ALGSTR_0:def 4
                      .= (ADDR).(a,x);
                      per cases;
                       suppose
A160:                   (ADDR).(x,c) <> x;
                        thus (a + b) + c = (addR(x,o)).(ab,c1) by Def8
                        .= addR(ab,c1) by Def5
                        .= x + cR by A135,A145,A160,Def4
                        .= addR(a1,bc) by A77,A159,A160,A158,Def4
                        .= (addR(x,o)).(a,bc) by Def5
                        .= a + (b + c) by Def8;
                       end;
                       suppose
A161:                   (ADDR).(x,c) = x;
                        thus (a + b) + c = (addR(x,o)).(ab,c1) by Def8
                        .= addR(ab,c1) by Def5
                        .= o by A135,A145,A161,Def4
                        .= addR(a1,bc) by A77,A159,A161,A158,Def4
                        .= (addR(x,o)).(a,bc) by Def5
                        .= a + (b + c) by Def8;
                       end;
                      end;
                     end;
                    end;
                    suppose
A162:                (ADDR).(a,b) <> x;
A163:                a + b = (addR(x,o)).(a1,b1) by Def8
                     .= addR(a1,b1) by Def5
                     .= aR + bR by A162,A130,A77,Def4; then
A164:                a + b <> o by A1; then
                     not a + b in {o} by TARSKI:def 1; then
A165:                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 A165;
                     per cases;
                      suppose
A166:                  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
A167:                    (ADDR).(b,x) <> x;
A168:                    b + c = (addR(x,o)).(b1,c1) by Def8
                         .= addR(b1,c1) by Def5
                         .= bR + x by A166,A130,A167,Def4; then
A169:                    b + c <> o by A1; then
                         not b + c in {o} by TARSKI:def 1;then
A170:                    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 A170;
A171:                    (ADDR).(ab,x)=(aR + bR) + x by A163
                         .= aR + (bR + x) by RLVECT_1:def 3
                         .= (the addF of F).(a,bc) by A168;
                         per cases;
                          suppose
A172:                      (ADDR).(ab,x) <> x;
                           thus (a + b) + c = (addR(x,o)).(ab,c1) by Def8
                           .= addR(ab,c1) by Def5
                           .= (aR + bR) + x by A1,A163,A166,A172,Def4
                           .= aR + (bR + x) by RLVECT_1:def 3
                           .= addR(a1,bc) by A77,A168,A169,A171,A172,Def4
                           .= (addR(x,o)).(a,bc) by Def5
                           .= a + (b + c) by Def8;
                          end;
                          suppose
A173:                      (ADDR).(ab,x) = x;
                           thus (a + b) + c = (addR(x,o)).(ab,c1) by Def8
                           .= addR(ab,c1) by Def5
                           .= o by A164,A166,A173,Def4
                           .= addR(a1,bc) by A77,A173,A169,A171,Def4
                           .= (addR(x,o)).(a,bc) by Def5
                           .= a + (b + c) by Def8;
                          end;
                         end;
                         suppose
A174:                     (ADDR).(b,x) = x;
A175:                     b + c =(addR(x,o)).(b1,c1) by Def8
                          .= addR(b1,c1) by Def5
                          .= o by A166,A130,A174,Def4; then
                          b + c in {o} by TARSKI:def 1; then
                        reconsider bc = b+c as Element of C by XBOOLE_0:def 3;
A176:                     (aR + bR) + x = aR + (bR + x) by RLVECT_1:def 3
                          .= (the addF of F).(a,x) by A174;
                          per cases;
                           suppose
A177:                       (ADDR).(ab,x) <> x;
                            thus (a + b) + c =  (addR(x,o)).(ab,c1) by Def8
                            .= addR(ab,c1) by Def5
                            .= (aR + bR) + x by A163,A1,A166,A177,Def4
                            .= addR(a1,bc) by A163,A77,A177,A175,A176,Def4
                            .= (addR(x,o)).(a,bc) by Def5
                            .= a + (b + c) by Def8;
                           end;
                           suppose
A178:                       (ADDR).(ab,x) = x;
                            thus (a + b) + c = (addR(x,o)).(ab,c1) by Def8
                             .= addR(ab,c1) by Def5
                             .= o by A164,A166,A178,Def4
                             .= addR(a1,bc) by A163,A77,A178,A175,A176,Def4
                             .= (addR(x,o)).(a,bc) by Def5
                             .= a + (b + c) by Def8;
                            end;
                           end;
                          end;
                          suppose
A179:                      c <> o; then
                           not c in {o} by TARSKI:def 1; then
A180:                      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 A180;
                           per cases;
                            suppose
A181:                        (ADDR).(b,c) <> x;
A182:                        b + c = (addR(x,o)).(b1,c1) by Def8
                             .= addR(b1,c1) by Def5
                             .= bR + cR by A179,A130,A181,Def4; then
A183:                        b + c <> o by A1; then
                             not b + c in {o} by TARSKI:def 1; then
A184:                        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 A184;
A185:                        (ADDR).(ab,c) = (aR + bR) + cR  by A163
                             .= aR + (bR + cR) by RLVECT_1:def 3
                             .= (ADDR).(a,bc) by A182;
                             per cases;
                              suppose
A186:                          (ADDR).(ab,c) <> x;
                               thus (a + b) + c = (addR(x,o)).(ab,c1) by Def8
                               .= addR(ab,c1) by Def5
                               .= (aR + bR) + cR by A163,A164,A179,A186,Def4
                               .= aR + (bR + cR) by RLVECT_1:def 3
                               .= addR(a1,bc) by A183,A77,A186,A182,A185,Def4
                               .= (addR(x,o)).(a,bc) by Def5
                               .= a + (b + c) by Def8;
                              end;
                              suppose
A187:                          (ADDR).(ab,c) = x;
                               thus (a + b) + c = (addR(x,o)).(ab,c1) by Def8
                               .= addR(ab,c1) by Def5
                               .= o by A164,A179,A187,Def4
                               .= addR(a1,bc) by A183,A77,A187,A185,Def4
                               .= (addR(x,o)).(a,bc) by Def5
                               .= a + (b + c) by Def8;
                              end;
                             end;
                             suppose
A188:                         (ADDR).(b,c) = x;
A189:                         b + c = (addR(x,o)).(b1,c1) by Def8
                              .= addR(b1,c1) by Def5
                              .= o by A130,A179,A188,Def4; then
                              b + c in {o} by TARSKI:def 1; then
                 reconsider bc = b+c as Element of C by XBOOLE_0:def 3;
A190:                         (aR + bR) + cR = aR + (bR + cR) by RLVECT_1:def 3
                              .= (ADDR).(a,x) by A188;
                              per cases;
                               suppose
A191:                           (ADDR).(ab,c) <> x;
                                thus
                                (a + b) + c = (addR(x,o)).(ab,c1) by Def8
                                .= addR(ab,c1) by Def5
                                .= (aR + bR) + cR by A163,A164,A179,A191,Def4
                                .= addR(a1,bc) by A77,A163,A189,A190,A191,Def4
                                .= (addR(x,o)).(a,bc) by Def5
                                .= a + (b + c) by Def8;
                               end;
                               suppose
A192:                           (ADDR).(ab,c) = x;
                                thus
                                (a + b) + c = (addR(x,o)).(ab,c1) by Def8
                                .= addR(ab,c1) by Def5
                                .= o by A164,A179,A192,Def4
                                .= addR(a1,bc) by A163,A77,A192,A189,A190,Def4
                                .= (addR(x,o)).(a,bc) by Def5
                                .= a + (b + c) by Def8;
                               end;
                              end;
                             end;
                            end;
                           end;
                          end;
                         end;
                         hence ExField(x,o) is add-associative
                         by RLVECT_1:def 3;
                        end;
