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

theorem Th11:
  for x being non trivial Element of F, o being object st not o in [#]F
  holds ExField(x,o) is distributive
  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);
A2:  [#]E = C by Def8;
A3:  now assume x + x = x; then
      x = 0.F by RLVECT_1:9;
      hence contradiction by Def2;
     end; then
     not x + x in {x} by TARSKI:def 1; then
     x + x in [#]F \ {x} by XBOOLE_0:def 5; then
     reconsider xpx = x + x as Element of C by XBOOLE_0:def 3;
A4:  x <> 0.F by Def2; then
A5:  x is left_mult-cancelable by ALGSTR_0:def 36;
A6:  now assume x * x = x; then
      x * x = x * 1.F;
      hence contradiction by A5,ALGSTR_0:def 20,Def2;
     end; then
     not x * x in {x} by TARSKI:def 1; then
     x * x in [#]F \ {x} by XBOOLE_0:def 5; then
     reconsider xmx = x * x as Element of C by XBOOLE_0:def 3;
A7: x * x <> o by A1;
A8:  now assume x * x + x = x; then
      x * x = 0.F by RLVECT_1:9; then
      x = 0.F by VECTSP_2:def 1;
      hence contradiction by Def2;
     end; then
     not x * x + x in {x} by TARSKI:def 1; then
     x * x + x in [#]F \ {x} by XBOOLE_0:def 5; then
     reconsider xmxpx = x * x + x as Element of C by XBOOLE_0:def 3;
A9:  now let a,b,c be Element of E;
      per cases;
       suppose
A10:   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
A11:     b = o; then
         b in {o} by TARSKI:def 1; then
         reconsider b1 = b as Element of C by XBOOLE_0:def 3;
A12:     a * b = (multR(x,o)).(a1,b1) by Def8
         .= multR(a1,b1) by Def7 .= x * x by A10,A11,A6,Def6;
         per cases;
          suppose
A13:       c = o; then
           c in {o} by TARSKI:def 1; then
           reconsider c1 = c as Element of C by XBOOLE_0:def 3;
A14:       b + c = (addR(x,o)).(b1,c1) by Def8
           .= addR(b1,c1) by Def5 .= x + x by A13,A11,A3,Def4;
A15:       a * c = (multR(x,o)).(a1,c1) by Def8 .= multR(a1,c1) by Def7
           .= x * x by A13,A10,A6,Def6;
A16:       (x * x) + (x * x) = x * (x + x) by VECTSP_1:def 2;
           per cases;
            suppose
A17:         (x * x) + (x * x) <> x;
A18:         xpx <> o & xmx <> o by A1;
             thus (a * b) + (a * c) = (addR(x,o)).(xmx,xmx) by A15,A12,Def8
             .= addR(xmx,xmx) by Def5 .= (x * x) + (x * x) by A18,A17,Def4
             .= multR(a1,xpx) by A1,A17,A10,A16,Def6
             .= (multR(x,o)).(a1,xpx) by Def7 .= a * (b + c) by A14,Def8;
            end;
            suppose
A19:         (x * x) + (x * x) = x;
A20:         xpx <> o & xmx <> o by A1;
             thus
             (a * b) + (a * c) = (addR(x,o)).(xmx,xmx) by A15,A12,Def8
             .= addR(xmx,xmx) by Def5 .= o by A20,A19,Def4
             .= multR(a1,xpx) by A20,A19,A10,A16,Def6
             .= (multR(x,o)).(a1,xpx) by Def7 .= a * (b + c) by A14,Def8;
            end;
          end;
          suppose
A21:       c <> o; then
           not c in {o} by TARSKI:def 1; then
A22:       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 A22;
           per cases;
            suppose
A23:         x + cR = x; then
A24:         cR = 0.F by RLVECT_1:9;
A25:         b + c = (addR(x,o)).(b1,c1) by Def8 .= addR(b1,c1) by Def5
             .= o by A23,A21,A11,Def4; then
             b + c in {o} by TARSKI:def 1; then
             reconsider bc1 = b + c as Element of C by XBOOLE_0:def 3;
A26:         x * cR <> x by A24,Def2;
A27:         a * c = (multR(x,o)).(a1,c1) by Def8
             .= multR(a1,c1) by Def7 .= x * cR by A26,A21,A10,Def6; then
A28:         a * c <> o by A1;
             reconsider ac1 = a * c as Element of C by Def8;
A29:         now assume  x * x + x * cR = x; then
             x * (x + cR) = x * 1.F by VECTSP_1:def 2; then
             x + cR = 1.F by A4,VECTSP_2:8;
             hence contradiction by A23,Def2;
            end;
            thus
            (a * b) + (a * c) = (addR(x,o)).(xmx,ac1) by A12,Def8
            .= addR(xmx,ac1) by Def5
            .= x * x + x * cR by A7,A28,A27,A29,Def4
            .= x * x by A23,VECTSP_1:def 2
            .= multR(a1,bc1) by A10,A25,A6,Def6
            .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
           end;
           suppose
A30:        x + cR <> x;
A31:        b + c = (addR(x,o)).(b1,c1) by Def8 .= addR(b1,c1) by Def5
            .= x + cR by A30,A21,A11,Def4; then
A32:        b + c <> o by A1; then
            not b + c in {o} by TARSKI:def 1; then
A33:        b + c in [#]F \ {x} by A2,XBOOLE_0:def 3;
            reconsider bc1 = b + c as Element of C by Def8;
            reconsider bcR = b + c as Element of F by A33;
            per cases;
             suppose
A34:          x * cR = x;
A35:          a * c = (multR(x,o)).(a1,c1) by Def8 .= multR(a1,c1) by Def7
              .= o by A34,A21,A10,Def6; then
              a * c in {o} by TARSKI:def 1; then
              reconsider ac1 = a * c as Element of C by XBOOLE_0:def 3;
A36:          x * (x + cR) <> x by A34,A8,VECTSP_1:def 2;
              thus
              (a * b) + (a * c) = (addR(x,o)).(xmx,ac1) by A12,Def8
              .= addR(xmx,ac1) by Def5 .= x * x + x by A8,A1,A35,Def4
              .= x * (x + cR) by A34,VECTSP_1:def 2
              .= multR(a1,bc1) by A10,A31,A1,A36,Def6
              .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
             end;
             suppose
A37:          x * cR <> x;
A38:          a * c = (multR(x,o)).(a1,c1) by Def8
              .= multR(a1,c1) by Def7 .= x * cR by A37,A21,A10,Def6; then
A39:          a * c <> o by A1; then
              not a * c in {o} by TARSKI:def 1; then
A40:          a * c in [#]F \ {x} by A2,XBOOLE_0:def 3;
              reconsider ac1 = a * c as Element of C by Def8;
              reconsider acR = a * c as Element of F by A40;
              per cases;
               suppose
A41:            x * x + x * cR <> x; then
A42:            x * (x + cR) <> x by VECTSP_1:def 2;
                thus
                (a * b) + (a * c) = (addR(x,o)).(xmx,ac1) by A12,Def8
                .= addR(xmx,ac1) by Def5
                .= x * x + x * cR by A41,A7,A38,A39,Def4
                .= x * (x + cR) by VECTSP_1:def 2
                .= multR(a1,bc1) by A42,A10,A31,A1,Def6
                .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
               end;
               suppose
A43:            x * x + x * cR = x; then
A44:            x * (x + cR) = x by VECTSP_1:def 2;
                thus
                (a * b) + (a * c) = (addR(x,o)).(xmx,ac1) by A12,Def8
                .= addR(xmx,ac1) by Def5 .= o by A43,A7,A38,A39,Def4
                .= multR(a1,bc1) by A44,A10,A31,A32,Def6
                .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
               end;
              end;
           end;
         end;
       end;
       suppose
A45:    b <> o; then
        not b in {o} by TARSKI:def 1; then
A46:    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 A46;
        per cases;
         suppose
A47:      x * bR = x;
A48:      a * b = (multR(x,o)).(a1,b1) by Def8 .= multR(a1,b1) by Def7
          .= o by A10,A45,A47,Def6; then
          a * b in {o} by TARSKI:def 1; then
          reconsider ab1 = a * b as Element of C by XBOOLE_0:def 3;
          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:        a * c = (multR(x,o)).(a1,c1) by Def8 .= multR(a1,c1) by Def7
            .= x * x by A10,A49,A6,Def6;
A51:        now assume bR + x = x; then
            bR = 0.F by RLVECT_1:9;
            hence contradiction by A47,Def2;
           end;
A52:       b + c = (addR(x,o)).(b1,c1) by Def8 .= addR(b1,c1) by Def5
           .= bR + x by A51,A49,A45,Def4; then
           b + c <> o by A1; then
           not b + c in {o} by TARSKI:def 1; then
A53:       b + c in [#]F \ {x} by A2,XBOOLE_0:def 3;
           reconsider bc1 = b + c as Element of C by Def8;
A54:       x * (bR + x) = x + x * x by A47,VECTSP_1:def 2;
           reconsider bcR = b + c as Element of F by A53;
           thus
           (a * b) + (a * c) = (addR(x,o)).(ab1,xmx) by A50,Def8
           .= addR(ab1,xmx) by Def5 .= x + x * x by A8,A48,A1,Def4
           .= multR(a1,bc1) by A10,A52,A8,A54,A1,Def6
           .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
          end;
          suppose
A55:       c <> o; then
           not c in {o} by TARSKI:def 1; then
A56:       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 A56;
           per cases;
            suppose
A57:         bR + cR = x;
A58:         b + c = (addR(x,o)).(b1,c1) by Def8 .= addR(b1,c1) by Def5
             .= o by A45,A55,A57,Def4; then
             b + c in {o} by TARSKI:def 1; then
             reconsider bc1 = b+c as Element of C by XBOOLE_0:def 3;
             per cases;
              suppose
A59:           x * cR = x;
A60:           a * c = (multR(x,o)).(a1,c1) by Def8 .= multR(a1,c1) by Def7
               .= o by A10,A55,A59,Def6; then
               a * c in {o} by TARSKI:def 1; then
               reconsider ac1 = a*c as Element of C by XBOOLE_0:def 3;
A61:           x * x = x + x
               proof
A62:            x <> 0.F by Def2;
                x * bR = x * 1.F by A47; then
A63:            bR = 1.F by A62,VECTSP_2:8;
A64:            x * cR = x * 1.F by A59;
A65:            x = 1.F + 1.F by A57,A62,A63,A64,VECTSP_2:8;
                hence x * x = 1.F * (1.F+1.F) + 1.F * (1.F+1.F)
                by VECTSP_1:def 3
                .= x + x by A65;
               end;
               thus
               (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
               .= addR(ab1,ac1) by Def5 .= x + x by A60,A48,A3,Def4
               .= multR(a1,bc1) by A61,A10,A58,A6,Def6
               .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
              end;
              suppose
A66:           x * cR <> x;
A67:           a * c = (multR(x,o)).(a1,c1) by Def8 .= multR(a1,c1) by Def7
               .= x * cR by A10,A55,A66,Def6; then
               a * c <> o by A1; then
               not a*c in {o} by TARSKI:def 1; then
A68:           a*c in [#]F \ {x} by A2,XBOOLE_0:def 3;
               reconsider ac1 = a*c as Element of C by Def8;
               reconsider acR = a*c as Element of F by A68;
A69:           x * x = x + x * cR by A47,A57,VECTSP_1:def 2;
               thus
               (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
               .= addR(ab1,ac1) by Def5
               .= x + x * cR by A1,A67,A48,A69,A6,Def4
               .= multR(a1,bc1) by A69,A10,A58,A6,Def6
               .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
              end;
             end;
             suppose
A70:          bR + cR <> x;
A71:          b + c = (addR(x,o)).(b1,c1) by Def8 .= addR(b1,c1) by Def5
              .= bR + cR by A45,A55,A70,Def4; then
A72:          b + c <> o by A1; then
              not b + c in {o} by TARSKI:def 1; then
A73:          b+c in [#]F \ {x} by A2,XBOOLE_0:def 3;
              reconsider bc1 = b+c as Element of C by Def8;
              reconsider bcR = b+c as Element of F by A73;
              per cases;
               suppose
A74:            x * cR = x;
A75:            a * c = (multR(x,o)).(a1,c1) by Def8 .= multR(a1,c1) by Def7
                .= o by A10,A55,A74,Def6; then
                a * c in {o} by TARSKI:def 1; then
                reconsider ac1 = a*c as Element of C by XBOOLE_0:def 3;
A76:            now assume
A77:             x * (bR + cR) = x;
A78:             x <> 0.F by Def2;
                 x * 1.F = x * 1.F + x * cR by A47,A77,VECTSP_1:def 2
                 .= x * (1.F + cR) by VECTSP_1:def 2; then
                 1.F = 1.F + cR by A78,VECTSP_2:8; then
                 cR = 0.F by RLVECT_1:9;
                 hence contradiction by A74,Def2;
                end;
                thus
                (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                .= addR(ab1,ac1) by Def5 .= x + x by A3,A75,A48,Def4
                .= x * (bR + cR) by A74,A47,VECTSP_1:def 2
                .= multR(a1,bc1) by A76,A10,A1,A71,Def6
                .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
               end;
               suppose
A79:            x * cR <> x;
A80:            a * c = (multR(x,o)).(a1,c1) by Def8 .= multR(a1,c1) by Def7
                .= x * cR by A10,A55,A79,Def6; then
A81:            a * c <> o by A1; then
                not a * c in {o} by TARSKI:def 1; then
A82:            a*c in [#]F \ {x} by A2,XBOOLE_0:def 3;
                reconsider ac1=a*c as Element of C by Def8;
                reconsider acR=a*c as Element of F by A82;
A83:            x * (bR + cR) = x + x * cR by A47,VECTSP_1:def 2;
                per cases;
                 suppose
A84:              x + x * cR = x;
                  thus (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                  .= addR(ab1,ac1) by Def5
                  .= o by A81,A80,A48,A84,Def4
                  .= multR(a1,bc1) by A84,A83,A10,A71,A72,Def6
                  .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
                 end;
                 suppose
A85:              x + x * cR <> x;
                  thus (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                  .= addR(ab1,ac1) by Def5
                  .= x + x * cR by A1,A80,A48,A85,Def4
                  .= multR(a1,bc1) by A85,A83,A10,A71,A1,Def6
                  .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
                 end;
                end;
               end;
              end;
             end;
             suppose
A86:          x * bR <> x;
A87:          a * b = (multR(x,o)).(a1,b1) by Def8 .= multR(a1,b1) by Def7
              .= x * bR by A10,A45,A86,Def6; then
A88:          a * b <> o by A1; then
              not a * b in {o} by TARSKI:def 1; then
A89:          a * b in [#]F \ {x} by A2,XBOOLE_0:def 3;
              reconsider ab1 = a * b as Element of C by Def8;
              reconsider abR = a * b as Element of F by A89;
              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;
A91:           a * c = (multR(x,o)).(a1,c1) by Def8 .= multR(a1,c1) by Def7
               .= x * x by A10,A90,A6,Def6; then
A92:           a * c <> o by A1; then
               not a * c in {o} by TARSKI:def 1; then
A93:           a * c in [#]F \ {x} by A2,XBOOLE_0:def 3;
               reconsider ac1 = a * c as Element of C by Def8;
               reconsider acR = a * c as Element of F by A93;
               per cases;
                suppose
A94:             bR + x = x;
A95:             b + c = (addR(x,o)).(b1,c1) by Def8 .= addR(b1,c1) by Def5
                 .= o by A45,A90,A94,Def4; then
                 b + c in {o} by TARSKI:def 1; then
                 reconsider bc1 = b+c as Element of C by XBOOLE_0:def 3;
A96:             x * bR + x * x = x * x by A94,VECTSP_1:def 2;
                 thus
                 (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                 .= addR(ab1,ac1) by Def5
                 .= x * bR + x * x by A87,A88,A91,A92,A96,A6,Def4
                 .= multR(a1,bc1) by A10,A95,A96,A6,Def6
                 .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
                end;
                suppose
A97:             bR + x <> x;
A98:             b + c = (addR(x,o)).(b1,c1) by Def8 .= addR(b1,c1) by Def5
                 .= bR + x by A45,A90,A97,Def4; then
A99:             b + c <> o by A1; then
                 not b + c in {o} by TARSKI:def 1; then
A100:            b+c in [#]F \ {x} by A2,XBOOLE_0:def 3;
                 reconsider bc1 = b+c as Element of C by Def8;
                 reconsider bcR = b+c as Element of F by A100;
                 per cases;
                  suppose
A101:              x * bR + x * x <> x;
A102:              x * bR + x * x = x * (bR + x) by VECTSP_1:def 2;
                   thus
                   (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                   .= addR(ab1,ac1) by Def5
                   .= x * bR + x * x by A87,A88,A91,A92,A101,Def4
                   .= multR(a1,bc1) by A10,A98,A1,A101,A102,Def6
                   .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
                  end;
                  suppose
A103:              x * bR + x * x = x;
A104:              x * bR + x * x = x * (bR + x) by VECTSP_1:def 2;
                   thus
                   (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                   .= addR(ab1,ac1) by Def5
                   .= o by A87,A88,A91,A92,A103,Def4
                   .= multR(a1,bc1) by A10,A98,A99,A103,A104,Def6
                   .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
                  end;
                 end;
                end;
                suppose
A105:            c <> o; then
                 not c in {o} by TARSKI:def 1; then
A106:            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 A106;
                 per cases;
                 suppose
A107:             bR + cR = x;
A108:             b + c = (addR(x,o)).(b1,c1) by Def8 .= addR(b1,c1) by Def5
                  .= o by A45,A105,A107,Def4; then
                  b + c in {o} by TARSKI:def 1; then
                  reconsider bc1 = b+c as Element of C by XBOOLE_0:def 3;
A109:             x * bR + x * cR = x * x by A107,VECTSP_1:def 2;
                  per cases;
                   suppose
A110:               x * cR <> x;
A111:               a * c = (multR(x,o)).(a1,c1) by Def8
                    .= multR(a1,c1) by Def7
                    .= x * cR by A10,A105,A110,Def6; then
A112:               a * c <> o by A1; then
                    not a * c in {o} by TARSKI:def 1; then
A113:               a*c in [#]F \ {x} by A2,XBOOLE_0:def 3;
                    reconsider ac1=a*c as Element of C by Def8;
                    reconsider acR=a*c as Element of F by A113;
                    thus
                    (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                    .= addR(ab1,ac1) by Def5
                    .= x * bR + x * cR by A87,A88,A111,A112,A109,A6,Def4
                    .= multR(a1,bc1) by A10,A108,A6,A109,Def6
                    .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
                   end;
                   suppose
A114:               x * cR = x;
A115:               a * c = (multR(x,o)).(a1,c1) by Def8
                    .= multR(a1,c1) by Def7
                    .= o by A10,A105,A114,Def6; then
                    a * c in {o} by TARSKI:def 1; then
                    reconsider ac1 = a*c as Element of C by XBOOLE_0:def 3;
A116:               now assume x * bR + x = x; then
A117:                x * 1.F = x * (bR + cR) by A114,VECTSP_1:def 2;
                     x <> 0.F by Def2; then
                     bR + cR = 1.F by A117,VECTSP_2:8;
                     hence contradiction by A107,Def2;
                    end;
                    thus
                    (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                    .= addR(ab1,ac1) by Def5
                    .= x * bR + x  by A87,A1,A115,A116,Def4
                    .= multR(a1,bc1) by A10,A108,A6,A109,A114,Def6
                    .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
                   end;
                  end;
                  suppose
A118:              bR + cR <> x;
A119:              b + c = (addR(x,o)).(b1,c1) by Def8 .= addR(b1,c1) by Def5
                   .= bR + cR by A45,A105,A118,Def4; 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 bc1 = b+c as Element of C by Def8;
                   reconsider bcR = b+c as Element of F by A121;
A122:              x * bR + x * cR = x * (bR + cR) by VECTSP_1:def 2;
                   per cases;
                    suppose
A123:               x * cR <> x;
A124:               a * c = (multR(x,o)).(a1,c1) by Def8
                    .= multR(a1,c1) by Def7
                    .= x * cR by A10,A105,A123,Def6; then
A125:               a * c <> o by A1; then
                    not a * c in {o} by TARSKI:def 1; then
A126:               a*c in [#]F \ {x} by A2,XBOOLE_0:def 3;
                    reconsider ac1=a*c as Element of C by Def8;
                    reconsider acR=a*c as Element of F by A126;
                    per cases;
                     suppose
A127:                 x * bR + x * cR <> x;
                      thus
                      (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                      .= addR(ab1,ac1) by Def5
                      .= x * bR + x * cR by A87,A88,A124,A125,A127,Def4
                      .= multR(a1,bc1) by A10,A119,A1,A127,A122,Def6
                      .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
                     end;
                     suppose
A128:                 x * bR + x * cR = x;
                      thus
                      (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                      .= addR(ab1,ac1) by Def5
                      .= o by A87,A88,A124,A125,A128,Def4
                      .= multR(a1,bc1) by A10,A119,A120,A128,A122,Def6
                      .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
                     end;
                    end;
                    suppose
A129:                x * cR = x;
A130:                a * c = (multR(x,o)).(a1,c1) by Def8
                     .= multR(a1,c1) by Def7
                     .= o by A10,A105,A129,Def6; then
                     a * c in {o} by TARSKI:def 1; then
                     reconsider ac1 = a*c as Element of C by XBOOLE_0:def 3;
A131:                x * bR + x = x * (bR + cR) by A129,VECTSP_1:def 2;
                     per cases;
                      suppose
A132:                  x * bR + x <> x;
                       thus
                       (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                       .= addR(ab1,ac1) by Def5
                       .= x * bR + x by A87,A1,A130,A132,Def4
                       .= multR(a1,bc1) by A10,A119,A1,A132,A131,Def6
                       .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
                      end;
                      suppose
A133:                  x * bR + x = x;
                       thus
                      (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                      .= addR(ab1,ac1) by Def5
                      .= o by A87,A88,A130,A133,Def4
                      .= multR(a1,bc1) by A10,A119,A120,A133,A131,Def6
                      .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
                     end;
                    end;
                   end;
                 end;
               end;
             end;
           end;
           suppose
A134:       a <> o; then
            not a in {o} by TARSKI:def 1; then
A135:       a in [#]F \ {x} by A2,XBOOLE_0:def 3;
            reconsider a1 = a as Element of carr(x,o) by Def8;
            reconsider aR = a as Element of [#]F by A135;
            per cases;
            suppose
A136:        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
A137:          aR * x = x;
A138:          a * b = (multR(x,o)).(a1,b1) by Def8 .= multR(a1,b1) by Def7
               .= o by A134,A136,A137,Def6; then
               a * b in {o} by TARSKI:def 1; then
               reconsider ab1 = a*b as Element of C by XBOOLE_0:def 3;
               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:            a * c = (multR(x,o)).(a1,c1) by Def8 .= multR(a1,c1) by Def7
                 .= o by A134,A139,A137,Def6; then
                 a * c in {o} by TARSKI:def 1; then
                 reconsider ac1 = a*c as Element of C by XBOOLE_0:def 3;
A141:            b + c = (addR(x,o)).(b1,c1) by Def8 .= addR(b1,c1) by Def5
                 .= x + x by A136,A139,A3,Def4; then
A142:            b + c <> o by A1;
                 reconsider bc1 = b+c as Element of C by Def8;
A143:            aR * (x + x) = x + x by A137,VECTSP_1:def 2;
                 thus
                 (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                 .= addR(ab1,ac1) by Def5 .= x + x by A140,A138,A3,Def4
                 .= multR(a1,bc1) by A143,A142,A134,A141,A3,Def6
                 .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
                end;
                suppose
A144:            c <> o; then
                 not c in {o} by TARSKI:def 1; then
A145:            c in [#]F \ {x} by A2,XBOOLE_0:def 3;
                 reconsider c1 = c as Element of carr(x,o) by Def8;
                 reconsider cR = c as Element of [#]F by A145;
A146:            now assume
A147:             aR * cR = x;
                  aR <> 0.F by A137,Def2; then
                  cR = x by A137,A147,VECTSP_2:8; then
                  cR in {x} by TARSKI:def 1;
                  hence contradiction by A145,XBOOLE_0:def 5;
                 end;
A148:            a * c = (multR(x,o)).(a1,c1) by Def8 .= multR(a1,c1) by Def7
                 .= aR * cR by A134,A144,A146,Def6; then
A149:            a * c <> o by A1;
                 reconsider ac1 = a*c as Element of C by Def8;
                 per cases;
                  suppose
A150:              x + cR = x; then
A151:              aR * x + aR * cR = aR * x by VECTSP_1:def 2;
A152:              b + c = (addR(x,o)).(b1,c1) by Def8 .= addR(b1,c1) by Def5
                   .= o by A144,A136,A150,Def4; then
                   b + c in {o} by TARSKI:def 1; then
                   reconsider bc1 = b+c as Element of C by XBOOLE_0:def 3;
                   thus
                   (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                   .= addR(ab1,ac1) by Def5
                   .= o by A137,A138,A148,A149,A151,Def4
                   .= multR(a1,bc1) by A134,A152,A137,Def6
                   .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
                  end;
                  suppose
A153:              x + cR <> x;
A154:              b + c = (addR(x,o)).(b1,c1) by Def8 .= addR(b1,c1) by Def5
                   .= x + cR by A136,A144,A153,Def4; then
A155:              b + c <> o by A1;
                   reconsider bc1 = b+c as Element of C by Def8;
A156:              now assume x + aR * cR = x; then
A158:               aR * x = aR * (x + cR) by A137,VECTSP_1:def 2;
                    aR <> 0.F by A137,Def2;
                    hence contradiction by A153,A158,VECTSP_2:8;
                   end;
A159:              x + aR * cR = aR * (x + cR) by A137,VECTSP_1:def 2;
                   thus
                   (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                   .= addR(ab1,ac1) by Def5
                   .= x + aR * cR by A138,A148,A1,A156,Def4
                   .= multR(a1,bc1) by A134,A154,A155,A156,A159,Def6
                   .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
                  end;
                 end;
                end;
                suppose
A160:            aR * x <> x;
A161:            a * b = (multR(x,o)).(a1,b1) by Def8  .= multR(a1,b1) by Def7
                 .= aR * x by A134,A136,A160,Def6; then
A162:            a * b <> o by A1; then
                 not a * b in {o} by TARSKI:def 1; then
A163:            a*b in [#]F \ {x} by A2,XBOOLE_0:def 3;
                 reconsider ab1 = a * b as Element of C by Def8;
                 reconsider abR = a * b as Element of F by A163;
                 per cases;
                  suppose
A164:              c = o; then
                   c in {o} by TARSKI:def 1; then
                   reconsider c1 = c as Element of C by XBOOLE_0:def 3;
A165:              a * c = (multR(x,o)).(a1,c1) by Def8 .= multR(a1,c1) by Def7
                   .= aR * x by A134,A164,A160,Def6; then
A166:              a * c <> o by A1;
                   reconsider ac1 = a*c as Element of C by Def8;
A167:              b + c = (addR(x,o)).(b1,c1) by Def8 .= addR(b1,c1) by Def5
                   .= x + x by A136,A164,A3,Def4; then
A168:              b + c <> o by A1;
                   reconsider bc1 = b+c as Element of C by Def8;
A169:              aR * (x + x) = aR * x + aR * x by VECTSP_1:def 2;
                   per cases;
                   suppose
A170:               aR * x + aR * x <> x;
                    thus
                    (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                    .= addR(ab1,ac1) by Def5
                    .= aR * x + aR * x by A161,A165,A166,A170,Def4
                    .= multR(a1,bc1) by A134,A170,A169,A168,A167,Def6
                    .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
                   end;
                   suppose
A171:               aR * x + aR * x = x;
                    thus
                    (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                    .= addR(ab1,ac1) by Def5 .= o by A161,A165,A166,A171,Def4
                    .= multR(a1,bc1) by A134,A171,A169,A168,A167,Def6
                    .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
                   end;
                  end;
                  suppose
A172:              c <> o; then
                   not c in {o} by TARSKI:def 1; then
A173:              c in [#]F \ {x} by A2,XBOOLE_0:def 3;
                   reconsider c1 = c as Element of carr(x,o) by Def8;
                   reconsider cR = c as Element of [#]F by A173;
                   per cases;
                   suppose
A174:              aR * cR = x;
A175:              a * c = (multR(x,o)).(a1,c1) by Def8 .= multR(a1,c1) by Def7
                   .= o by A134,A172,A174,Def6; then
                   a * c in {o} by TARSKI:def 1; then
                   reconsider ac1 = a*c as Element of C by XBOOLE_0:def 3;
                per cases;
                 suppose
A176:             x + cR = x;
A177:             b + c = (addR(x,o)).(b1,c1) by Def8 .= addR(b1,c1) by Def5
                  .= o by A136,A172,A176,Def4; then
                  b+c in {o} by TARSKI:def 1; then
                  reconsider bc1 = b+c as Element of C by XBOOLE_0:def 3;
A178:             aR * x + x = aR * x by A176,A174,VECTSP_1:def 2;
                  thus
                  (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                  .= addR(ab1,ac1) by Def5
                  .= aR * x + x by A160,A161,A1,A175,A178,Def4
                  .= multR(a1,bc1) by A134,A177,A178,A160,Def6
                  .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
                 end;
                 suppose
A179:             x + cR <> x;
A180:             b + c = (addR(x,o)).(b1,c1) by Def8 .= addR(b1,c1) by Def5
                  .= x + cR by A136,A172,A179,Def4; then
A181:             b + c <> o by A1;
                  reconsider bc1 = b+c as Element of C by Def8;
A182:             aR * x + x = aR * (x + cR) by A174,VECTSP_1:def 2;
                  per cases;
                   suppose
A183:              aR * x + x <> x;
                    thus
                    (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                    .= addR(ab1,ac1) by Def5
                    .= aR * x + x by A161,A1,A175,A183,Def4
                    .= multR(a1,bc1) by A134,A180,A181,A183,A182,Def6
                    .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
                   end;
                   suppose
A184:               aR * x + x = x;
                    thus
                    (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                    .= addR(ab1,ac1) by Def5
                    .= o by A161,A162,A175,A184,Def4
                    .= multR(a1,bc1) by A134,A180,A181,A184,A182,Def6
                    .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
                   end;
                 end;
               end;
               suppose
A185:           aR * cR <> x;
A186:           a * c = (multR(x,o)).(a1,c1) by Def8 .= multR(a1,c1) by Def7
                .= aR * cR by A134,A172,A185,Def6; then
A187:           a * c <> o by A1;
                reconsider ac1 = a*c as Element of C by Def8;
                per cases;
                 suppose
A188:             x + cR = x;
A189:             b + c = (addR(x,o)).(b1,c1) by Def8 .= addR(b1,c1) by Def5
                  .= o by A136,A172,A188,Def4; then
                  b+c in {o} by TARSKI:def 1; then
                  reconsider bc1 = b+c as Element of C by XBOOLE_0:def 3;
A190:             aR * x + aR * cR = aR * x by A188,VECTSP_1:def 2;
                  thus
                  (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                  .= addR(ab1,ac1) by Def5
                  .= aR * x + aR * cR by A160,A161,A162,A186,A187,A190,Def4
                  .= multR(a1,bc1) by A160,A134,A189,A190,Def6
                  .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
                 end;
                 suppose
A191:             x + cR <> x;
A192:             b + c = (addR(x,o)).(b1,c1) by Def8 .= addR(b1,c1) by Def5
                  .= x + cR by A136,A172,A191,Def4; then
A193:             b + c <> o by A1;
                  reconsider bc1 = b+c as Element of C by Def8;
A194:             aR * x + aR * cR = aR * (x + cR) by VECTSP_1:def 2;
                  per cases;
                   suppose
A195:               aR * x + aR * cR = x;
                    thus
                    (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                    .= addR(ab1,ac1) by Def5
                    .= o by A161,A162,A186,A187,A195,Def4
                    .= multR(a1,bc1) by A134,A192,A193,A195,A194,Def6
                    .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
                   end;
                   suppose
A196:               aR * x + aR * cR <> x;
                    thus
                    (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                    .= addR(ab1,ac1) by Def5
                    .= aR * x + aR * cR by A161,A162,A186,A187,A196,Def4
                    .= multR(a1,bc1) by A134,A192,A193,A196,A194,Def6
                    .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
                   end;
                 end;
               end;
             end;
           end;
         end;
         suppose
A197:     b <> o; then
          not b in {o} by TARSKI:def 1; then
A198:     b in [#]F \ {x} by A2,XBOOLE_0:def 3;
          reconsider b1 = b as Element of carr(x,o) by Def8;
          reconsider bR = b as Element of [#]F by A198;
          per cases;
           suppose
A199:       aR * bR = x;
A200:       a * b = (multR(x,o)).(a1,b1) by Def8 .= multR(a1,b1) by Def7
            .= o by A134,A197,A199,Def6; then
            a * b in {o} by TARSKI:def 1; then
            reconsider ab1 = a * b as Element of C by XBOOLE_0:def 3;
A201:       aR * (bR + x) = x + aR * x by A199,VECTSP_1:def 2;
A202:       now assume bR + x = x; then
             bR = 0.F by RLVECT_1:9;
             hence contradiction by A199,Def2;
            end;
A203:       now assume aR * x = x; then
A205:        x * aR = x * 1.F by GROUP_1:def 12;
             x <> 0.F by Def2; then
             aR = 1.F by A205,VECTSP_2:8; then
             bR in {x} by A199,TARSKI:def 1;
             hence contradiction by A198,XBOOLE_0:def 5;
            end;
            per cases;
             suppose
A206:         c = o; then
              c in {o} by TARSKI:def 1; then
              reconsider c1 = c as Element of C by XBOOLE_0:def 3;
A207:         b + c = (addR(x,o)).(b1,c1) by Def8 .= addR(b1,c1) by Def5
              .= bR + x by A197,A206,A202,Def4; then
A208:         b + c <> o by A1;
              reconsider bc1 = b+c as Element of C by Def8;
              reconsider bcR = b+c as Element of F by A207;
A209:         a * c = (multR(x,o)).(a1,c1) by Def8 .= multR(a1,c1) by Def7
              .= aR * x by A134,A206,A203,Def6; then
A210:         a * c <> o by A1; then
              not a * c in {o} by TARSKI:def 1; then
A211:         a*c in [#]F \ {x} by A2,XBOOLE_0:def 3;
              reconsider ac1 = a*c as Element of C by Def8;
              reconsider acR = a*c as Element of F by A211;
              per cases;
               suppose
A212:           x + aR * x = x;
                thus
                (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                .= addR(ab1,ac1) by Def5 .= o by A200,A209,A210,A212,Def4
                .= multR(a1,bc1) by A201,A134,A207,A208,A212,Def6
                .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
               end;
               suppose
A213:           x + aR * x <> x;
                thus (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                .= addR(ab1,ac1) by Def5
                .= x + aR * x by A200,A209,A1,A213,Def4
                .= multR(a1,bc1) by A201,A134,A207,A208,A213,Def6
                .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
               end;
              end;
              suppose
A214:          c <> o; then
               not c in {o} by TARSKI:def 1; then
A215:          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 A215;
               per cases;
                suppose
A216:           bR + cR = x;
A217:           b + c = (addR(x,o)).(b1,c1) by Def8 .= addR(b1,c1) by Def5
                .= o by A197,A214,A216,Def4; then
                b + c in {o} by TARSKI:def 1; then
                reconsider bc1 = b+c as Element of C by XBOOLE_0:def 3;
                per cases;
                 suppose
A218:             aR * cR = x;
A219:             a * c = (multR(x,o)).(a1,c1) by Def8
                  .= multR(a1,c1) by Def7
                  .= o by A134,A214,A218,Def6; then
                  a * c in {o} by TARSKI:def 1; then
                  reconsider ac1 = a * c as Element of C by XBOOLE_0:def 3;
                  thus
                  (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                  .= addR(ab1,ac1) by Def5 .= x + x by A200,A219,A3,Def4
                  .= aR * x by A216,A218,A199,VECTSP_1:def 2
                  .= multR(a1,bc1) by A134,A217,A203,Def6
                  .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
                 end;
                 suppose
A220:             aR * cR <> x;
A221:             a * c = (multR(x,o)).(a1,c1) by Def8 .= multR(a1,c1) by Def7
                  .= aR * cR by A134,A214,A220,Def6; then
                  not a * c = o by A1; then
                  not a * c in {o} by TARSKI:def 1; then
A222:             a*c in [#]F \ {x} by A2,XBOOLE_0:def 3;
                  reconsider ac1 = a * c as Element of C by Def8;
                  reconsider acR = a * c as Element of F by A222;
A223:             x + aR * cR = aR * x by A216,A199,VECTSP_1:def 2;
                  per cases;
                   suppose x + aR * cR = x;
                    hence (a * b)+(a * c) = a * (b + c)
                      by A203,A216,A199,VECTSP_1:def 2;
                   end;
                   suppose
A225:               x + aR * cR <> x;
                    thus (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                    .= addR(ab1,ac1) by Def5
                    .= x + aR * cR by A200,A221,A1,A225,Def4
                    .= multR(a1,bc1) by A134,A217,A225,A223,Def6
                    .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
                   end;
                  end;
                 end;
                 suppose
A226:             bR + cR <> x;
A227:             b + c = (addR(x,o)).(b1,c1) by Def8 .= addR(b1,c1) by Def5
                  .= bR + cR by A197,A214,A226,Def4; then
A228:             b + c <> o by A1;
                  reconsider bc1 = b+c as Element of C by Def8;
                  per cases;
                   suppose
A229:               aR * cR = x;
A230:               a * c = (multR(x,o)).(a1,c1) by Def8
                    .= multR(a1,c1) by Def7 .= o by A134,A214,A229,Def6; then
                    a * c in {o} by TARSKI:def 1; then
                    reconsider ac1 = a * c as Element of C by XBOOLE_0:def 3;
A231:               aR * (bR + cR) = x + x by A229,A199,VECTSP_1:def 2;
                    thus
                    (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                    .= addR(ab1,ac1) by Def5 .= x + x by A200,A230,A3,Def4
                    .= multR(a1,bc1) by A3,A231,A134,A228,A227,Def6
                    .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
                   end;
                   suppose
A232:               aR * cR <> x;
A233:               a * c = (multR(x,o)).(a1,c1) by Def8
                    .= multR(a1,c1) by Def7
                    .= aR * cR by A134,A214,A232,Def6; then
A234:               not a * c = o by A1;
                    reconsider ac1 = a * c as Element of C by Def8;
A235:               x + aR * cR = aR * (bR + cR) by A199,VECTSP_1:def 2;
                    per cases;
                     suppose
A236:                 x + aR * cR = x;
                      thus
                      (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                      .= addR(ab1,ac1) by Def5
                      .= o by A200,A233,A234,A236,Def4
                      .= multR(a1,bc1) by A134,A227,A228,A236,A235,Def6
                      .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
                     end;
                     suppose
A237:                 x + aR * cR <> x;
                      thus
                      (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                      .= addR(ab1,ac1) by Def5
                      .= x + aR * cR by A200,A233,A1,A237,Def4
                      .= multR(a1,bc1) by A134,A227,A228,A237,A235,Def6
                      .= (multR(x,o)).(a1,bc1) by Def7 .= a * (b + c) by Def8;
                     end;
                    end;
                   end;
                  end;
                 end;
                 suppose
A238:             aR * bR <> x;
A239:             a * b = (multR(x,o)).(a1,b1) by Def8
                  .= multR(a1,b1) by Def7
                  .= aR * bR by A134,A197,A238,Def6; then
A240:             a * b <> o by A1;
                  reconsider ab1 = a * b as Element of C by Def8;
                  reconsider abR = a * b as Element of F by A239;
                  per cases;
                   suppose
A241:               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
A242:                 bR + x <> x;
A243:                 b + c = (addR(x,o)).(b1,c1) by Def8
                      .= addR(b1,c1) by Def5
                      .= bR + x by A197,A241,A242,Def4; then
A244:                 b + c <> o by A1;
                      reconsider bc1 = b+c as Element of C by Def8;
                      per cases;
                       suppose
A245:                   aR * x <> x;
A246:                   a * c = (multR(x,o)).(a1,c1) by Def8
                        .= multR(a1,c1) by Def7
                        .= aR * x by A134,A241,A245,Def6; then
A247:                   a * c <> o by A1;
                        reconsider ac1 = a * c as Element of C by Def8;
A248:                   aR * bR + aR * x = aR * (bR + x) by VECTSP_1:def 2;
                        per cases;
                         suppose
A249:                     aR * bR + aR * x <> x;
                          thus
                          (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                          .= addR(ab1,ac1) by Def5
                          .= aR * bR + aR * x by A240,A239,A246,A247,A249,Def4
                          .= multR(a1,bc1) by A134,A243,A244,A249,A248,Def6
                          .= (multR(x,o)).(a1,bc1) by Def7
                          .= a * (b + c) by Def8;
                         end;
                         suppose
A250:                     aR * bR + aR * x = x;
                          thus
                          (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                          .= addR(ab1,ac1) by Def5
                          .= o by A240,A239,A246,A247,A250,Def4
                          .= multR(a1,bc1) by A134,A243,A244,A250,A248,Def6
                          .= (multR(x,o)).(a1,bc1) by Def7
                         .= a * (b + c) by Def8;
                         end;
                        end;
                        suppose
A251:                    aR * x = x;
A252:                    a * c = (multR(x,o)).(a1,c1) by Def8
                         .= multR(a1,c1) by Def7
                         .= o by A134,A241,A251,Def6; then
                         a * c in {o} by TARSKI:def 1; then
                     reconsider ac1 = a * c as Element of C by XBOOLE_0:def 3;
A253:                    aR * bR + x = aR * (bR + x) by A251,VECTSP_1:def 2;
                         per cases;
                          suppose
A254:                      aR * bR + x <> x;
                           thus
                           (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                           .= addR(ab1,ac1) by Def5
                           .= aR * bR + x by A1,A239,A252,A254,Def4
                           .= multR(a1,bc1) by A134,A243,A244,A254,A253,Def6
                           .= (multR(x,o)).(a1,bc1) by Def7
                           .= a * (b + c) by Def8;
                          end;
                          suppose
A255:                     aR * bR + x = x;
                           thus
                           (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                           .= addR(ab1,ac1) by Def5
                           .= o by A240,A239,A252,A255,Def4
                           .= multR(a1,bc1) by A134,A243,A244,A255,A253,Def6
                           .= (multR(x,o)).(a1,bc1) by Def7
                           .= a * (b + c) by Def8;
                          end;
                         end;
                        end;
                        suppose
A256:                    bR + x = x;
A257:                    b + c = (addR(x,o)).(b1,c1) by Def8
                         .= addR(b1,c1) by Def5
                         .= o by A197,A241,A256,Def4; then
                         b + c in {o} by TARSKI:def 1; then
                       reconsider bc1 = b+c as Element of C by XBOOLE_0:def 3;
                         per cases;
                          suppose
A258:                      aR * x <> x;
A259:                      a * c = (multR(x,o)).(a1,c1) by Def8
                           .= multR(a1,c1) by Def7
                           .= aR * x by A134,A241,A258,Def6; then
A260:                      a * c <> o by A1;
                           reconsider ac1 = a * c as Element of C by Def8;
A261:                      aR * bR + aR * x = aR * x by A256,VECTSP_1:def 2;
                           thus
                           (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                           .= addR(ab1,ac1) by Def5
                           .= aR * x by A240,A239,A259,A260,A258,A261,Def4
                           .= multR(a1,bc1) by A258,A134,A257,Def6
                           .= (multR(x,o)).(a1,bc1) by Def7
                           .= a * (b + c) by Def8;
                          end;
                          suppose
A262:                      aR * x = x;
A263:                      a * c = (multR(x,o)).(a1,c1) by Def8
                           .= multR(a1,c1) by Def7
                           .= o by A134,A241,A262,Def6; then
                           a * c in {o} by TARSKI:def 1; then
                     reconsider ac1 = a * c as Element of C by XBOOLE_0:def 3;
A264:                      aR * bR + x = x by A256,A262,VECTSP_1:def 2;
                           thus
                           (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                           .= addR(ab1,ac1) by Def5
                           .= o by A240,A239,A263,A264,Def4
                           .= multR(a1,bc1) by A262,A134,A257,Def6
                           .= (multR(x,o)).(a1,bc1) by Def7
                           .= a * (b + c) by Def8;
                          end;
                         end;
                        end;
                        suppose
A265:                    c <> o; then
                         not c in {o} by TARSKI:def 1; then
A266:                    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 A266;
A267:                    aR * bR + aR * cR = aR * (bR + cR) by VECTSP_1:def 2;
                         per cases;
                          suppose
A268:                      bR + cR <> x;
A269:                      b + c = (addR(x,o)).(b1,c1) by Def8
                           .= addR(b1,c1) by Def5
                           .= bR + cR by A197,A265,A268,Def4; then
A270:                      b + c <> o by A1;
                           reconsider bc1 = b+c as Element of C by Def8;
                           per cases;
                            suppose
A271:                        aR * cR <> x;
A272:                        a * c = (multR(x,o)).(a1,c1) by Def8
                             .= multR(a1,c1) by Def7
                             .= aR * cR by A134,A265,A271,Def6; then
A273:                        a * c <> o by A1;
                             reconsider ac1 = a * c as Element of C by Def8;
                             per cases;
                              suppose
A274:                          aR * bR + aR * cR <> x;
                               thus
                               (a * b) + (a * c) = (addR(x,o)).(ab1,ac1)
                               by Def8
                               .= addR(ab1,ac1) by Def5
                               .= aR * bR + aR * cR
                                by A240,A239,A272,A273,A274,Def4
                               .= multR(a1,bc1)
                                by A134,A269,A270,A274,A267,Def6
                               .= (multR(x,o)).(a1,bc1) by Def7
                               .= a * (b + c) by Def8;
                              end;
                              suppose
A275:                          aR * bR + aR * cR = x;
                               thus
                              (a * b) + (a * c) = (addR(x,o)).(ab1,ac1) by Def8
                              .= addR(ab1,ac1) by Def5
                              .= o by A240,A239,A272,A273,A275,Def4
                              .= multR(a1,bc1) by A134,A269,A270,A275,A267,Def6
                              .= (multR(x,o)).(a1,bc1) by Def7
                              .= a * (b + c) by Def8;
                             end;
                            end;
                            suppose
A276:                        aR * cR = x;
A277:                        a * c = (multR(x,o)).(a1,c1) by Def8
                             .= multR(a1,c1) by Def7
                             .= o by A134,A265,A276,Def6; then
                             a * c in {o} by TARSKI:def 1; then
                      reconsider ac1 = a * c as Element of C by XBOOLE_0:def 3;
                             per cases;
                              suppose
A278:                          aR * bR + x <> x; then
A279:                          aR * (bR + cR) <> x by A276,VECTSP_1:def 2;
                               thus
                               (a * b) + (a * c) = (addR(x,o)).(ab1,ac1)
                               by Def8
                               .= addR(ab1,ac1) by Def5
                               .= aR * bR + x by A1,A239,A277,A278,Def4
                               .= aR * (bR + cR) by A276,VECTSP_1:def 2
                               .= multR(a1,bc1) by A134,A269,A270,A279,Def6
                               .= (multR(x,o)).(a1,bc1) by Def7
                               .= a * (b + c) by Def8;
                              end;
                              suppose
A280:                          aR * bR + x = x;
                               thus
                               (a * b) + (a * c) = (addR(x,o)).(ab1,ac1)
                               by Def8
                               .= addR(ab1,ac1) by Def5
                               .= o by A240,A239,A277,A280,Def4
                               .= multR(a1,bc1)
                                  by A267,A276,A134,A269,A270,A280,Def6
                               .= (multR(x,o)).(a1,bc1) by Def7
                               .= a * (b + c) by Def8;
                              end;
                             end;
                            end;
                            suppose
A281:                        bR + cR = x;
A282:                        b + c = (addR(x,o)).(b1,c1) by Def8
                             .= addR(b1,c1) by Def5
                             .= o by A197,A265,A281,Def4; then
                             b + c in {o} by TARSKI:def 1; then
                        reconsider bc1 = b+c as Element of C by XBOOLE_0:def 3;
                             per cases;
                              suppose
A283:                         aR * cR <> x;
A284:                         a * c = (multR(x,o)).(a1,c1) by Def8
                              .= multR(a1,c1) by Def7
                              .= aR * cR by A134,A265,A283,Def6; then
A285:                         a * c <> o by A1;
                              reconsider ac1 = a * c as Element of C by Def8;
                              per cases;
                               suppose
A286:                           aR * bR + aR * cR <> x;
                                thus
                                (a * b) + (a * c)= (addR(x,o)).(ab1,ac1)
                                by Def8
                                .= addR(ab1,ac1) by Def5
                                .= aR * bR + aR * cR
                                   by A240,A239,A284,A285,A286,Def4
                                .= multR(a1,bc1)
                                   by A286,A134,A281,A282,A267,Def6
                                .= (multR(x,o)).(a1,bc1) by Def7
                                .= a * (b + c) by Def8;
                               end;
                               suppose
A287:                           aR * bR + aR * cR = x;
                                thus
                                (a * b) + (a * c) = (addR(x,o)).(ab1,ac1)
                                by Def8
                                .= addR(ab1,ac1) by Def5
                                .= o by A240,A239,A284,A285,A287,Def4
                                .= multR(a1,bc1)
                                   by A287,A134,A281,A282,A267,Def6
                                .= (multR(x,o)).(a1,bc1) by Def7
                                .= a * (b + c) by Def8;
                               end;
                             end;
                             suppose
A288:                         aR * cR = x;
A289:                         a * c = (multR(x,o)).(a1,c1) by Def8
                              .= multR(a1,c1) by Def7
                              .= o by A134,A265,A288,Def6; then
                              a * c in {o} by TARSKI:def 1; then
                     reconsider ac1 = a * c as Element of C by XBOOLE_0:def 3;
A290:                         aR * bR + x =aR * (bR + cR)
                              by A288,VECTSP_1:def 2;
                              per cases;
                               suppose
A291:                           aR * bR + x <> x;
                                thus
                                (a * b) + (a * c)= (addR(x,o)).(ab1,ac1)
                                 by Def8
                                .= addR(ab1,ac1) by Def5
                                .= aR * bR + x by A1,A239,A289,A291,Def4
                                .= multR(a1,bc1)
                                   by A291,A134,A281,A282,A290,Def6
                                .= (multR(x,o)).(a1,bc1) by Def7
                                .= a * (b + c) by Def8;
                               end;
                               suppose
A292:                           aR * bR + x = x;
                                thus
                                (a * b) + (a * c) = (addR(x,o)).(ab1,ac1)
                                by Def8
                                .= addR(ab1,ac1) by Def5
                                .= o by A240,A239,A289,A292,Def4
                                .= multR(a1,bc1)
                                   by A292,A134,A281,A282,A290,Def6
                                .= (multR(x,o)).(a1,bc1) by Def7
                                .= a * (b + c) by Def8;
                               end;
                              end;
                             end;
                            end;
                           end;
                          end;
                         end;
                        end;
                        now let a,b,c be Element of ExField(x,o);
                         thus
                         a * (b + c) = (a * b) + (a * c) by A9;
                         thus
                         (b + c) * a = a * (b + c) by GROUP_1:def 12
                         .= (a * b) + (a * c) by A9
                         .= (b * a) + (a * c) by GROUP_1:def 12
                         .= (b * a) + (c * a) by GROUP_1:def 12;
                        end;
                        hence ExField(x,o) is distributive;
                       end;
