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

theorem Th7:
   for x being non trivial Element of F, o being object st
   not o in [#]F holds ExField(x,o) is right_zeroed right_complementable
   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);
    set ADDR = the addF of F;
    consider xi being Element of F such that
A2: x+xi = 0.F by ALGSTR_0:def 11;
A3: [#]ExField(x,o) = C by Def8;
    o in {o} by TARSKI:def 1; then
    reconsider u1 = o as Element of C by XBOOLE_0:def 3;
    reconsider u = u1 as Element of ExField(x,o) by Def8;
    now let a be Element of ExField(x,o);
A4:  0.(ExField(x,o)) = 0.F by Def8;
     0.F <> x by Def2; then
     not 0.F in {x} by TARSKI:def 1; then
     0.F in [#]F \ {x} by XBOOLE_0:def 5; then
     reconsider u = 0.F as Element of C by XBOOLE_0:def 3;
A5:  o <> u by a1;
     per cases;
      suppose
A6:    a = o; then
       a in {o} by TARSKI:def 1; then
       reconsider a1 = a as Element of C by XBOOLE_0:def 3;
A7:    (the addF of F).(x,0.F) = x + 0.F .= x;
       thus
       a + 0.(ExField(x,o)) = (addR(x,o)).(a1,u) by A4,Def8
       .= addR(a1,u) by Def5 .= a by A5,A6,A7,Def4;
      end;
      suppose
A8:    a <> o; then
       not a in {o} by TARSKI:def 1; then
A9:    a in [#]F \ {x} by A3,XBOOLE_0:def 3;
       reconsider a1 = a as Element of C by Def8;
       reconsider aR = a as Element of [#]F by A9;
A10:   (the addF of F).(a,u) = aR + 0.F .= aR;
       not aR in {x} by A9,XBOOLE_0:def 5; then
A11:   (the addF of F).(a,u) <> x by A10,TARSKI:def 1;
       thus
       a + 0.(ExField(x,o)) = (addR(x,o)).(a1,u) by A4,Def8
       .= addR(a1,u) by Def5  .= aR + 0.F by A8,A5,A11,Def4 .= a;
      end;
     end;
     hence ExField(x,o) is right_zeroed by RLVECT_1:def 4;
     now let a be Element of ExField(x,o);
      per cases;
       suppose
A12:    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
A13:      xi = x; then
A14:      (the addF of F).(x,x) <> x by A2,Def2;
          a + u = (addR(x,o)).(a1,u1) by Def8 .= addR(a1,u1) by Def5
          .= (the addF of F).(x,xi) by A12,A13,A14,Def4
          .= 0.(ExField(x,o)) by A2,Def8;
          hence a is right_complementable by ALGSTR_0:def 11;
         end;
         suppose xi <> x; then
          not xi in {x} by TARSKI:def 1; then
          xi in [#]F \ {x} by XBOOLE_0:def 5; then
          reconsider x1i = xi as Element of carr(x,o) by XBOOLE_0:def 3;
          reconsider b = x1i as Element of ExField(x,o) by Def8;
A15:      (the addF of F).(x,b) <> x by A2,Def2;
          a + b = (addR(x,o)).(a1,x1i) by Def8 .= addR(a1,x1i) by Def5
          .= (the addF of F).(x,xi) by A1,A12,A15,Def4
          .= 0.(ExField(x,o)) by A2,Def8;
          hence a is right_complementable by ALGSTR_0:def 11;
         end;
        end;
        suppose
A16:     a <> o; then
         not a in {o} by TARSKI:def 1; then
A17:     a in [#]F \ {x} by A3,XBOOLE_0:def 3;
         reconsider a1 = a as Element of C by Def8;
         reconsider aR = a as Element of [#]F by A17;
         consider aRi being Element of F such that
A18:     aR + aRi = 0.F by ALGSTR_0:def 11;
         per cases;
          suppose
A19:       aRi = x; then
A20:       (the addF of F).(a,x) <> x by A18,Def2;
           a + u = (addR(x,o)).(a1,u1) by Def8 .= addR(a1,u1) by Def5
           .= (the addF of F).(aR,aRi) by A16,A19,A20,Def4
           .= 0.(ExField(x,o)) by A18,Def8;
           hence a is right_complementable by ALGSTR_0:def 11;
          end;
          suppose aRi <> x; then
           not aRi in {x} by TARSKI:def 1; then
           aRi in [#]F \ {x} by XBOOLE_0:def 5; then
           reconsider a1i = aRi as Element of C by XBOOLE_0:def 3;
           reconsider b = a1i as Element of ExField(x,o) by Def8;
A21:       (the addF of F).(a,b) <> x by A18,Def2;
A22:       aR <> o & aRi <> o by a1;
           a + b = (addR(x,o)).(a1,aRi) by Def8 .= addR(a1,a1i) by Def5
           .= (the addF of F).(aR,aRi) by A21,A22,Def4
           .= 0.(ExField(x,o)) by A18,Def8;
           hence a is right_complementable by ALGSTR_0:def 11;
          end;
         end;
        end;
        hence ExField(x,o) is right_complementable by ALGSTR_0:def 16;
       end;
