reserve V for RealLinearSpace;
reserve u,u1,u2,v,v1,v2,w,w1,y for VECTOR of V;
reserve a,a1,a2,b,b1,b2,c1,c2 for Real;
reserve x,z for set;
reserve p,p1,q,q1 for Element of Lambda(OASpace(V));
reserve POS for non empty ParOrtStr;
reserve p,p1,p2,q,q1,r,r1,r2 for Element of AMSpace(V,w,y);

theorem
  for POS being OrtAfSp st the AffinStruct of POS is AffinPlane
     holds POS is OrtAfPl
proof
  let POS be OrtAfSp such that
A1: the AffinStruct of POS is AffinPlane;
A2: now
    let a,b,c,d,p,q,r,s be Element of POS;
    thus (a,b _|_ a,b implies a=b) & a,b _|_ c,c & (a,b _|_ c,d implies a,b
_|_ d,c & c,d _|_ a,b) & (a,b _|_ p,q & a,b // r,s implies p,q _|_ r,s or a=b)
    by Def7;
    thus a,b _|_ p,q & a,b _|_ r,s implies p,q // r,s or a=b
    proof
      reconsider a9=a,b9=b,p9=p,q9=q,r9=r,s9=s as
      Element of the AffinStruct of POS;
      assume that
A3:   a,b _|_ p,q and
A4:   a,b _|_ r,s;
A5:   p,q _|_ a,b by A3,Def7;
A6:   r,s _|_ a,b by A4,Def7;
      assume
A7:   not thesis;
      then
A8:   not p9,q9 // r9,s9 by Th36;
      then
A9:   p9<>q9 by AFF_1:3;
      consider x9 being Element of the AffinStruct of POS such that
A10:  LIN p9,q9,x9 and
A11:  LIN r9,s9,x9 by A1,A8,AFF_1:60;
      reconsider x=x9 as Element of POS;
A12:  r9<>s9 by A8,AFF_1:3;
      LIN s9,r9,x9 by A11,AFF_1:6;
      then s9,r9 // s9,x9 by AFF_1:def 1;
      then
A13:  r9,s9 // x9,s9 by AFF_1:4;
      then r,s // x,s by Th36;
      then a,b _|_ x,s by A12,A6,Def7;
      then
A14:  x,s _|_ a,b by Def7;
      LIN q9,p9,x9 by A10,AFF_1:6;
      then q9,p9 // q9,x9 by AFF_1:def 1;
      then p9,q9 // x9,q9 by AFF_1:4;
      then p,q // x,q by Th36;
      then
A15:  a,b _|_ x,q by A9,A5,Def7;
A16:  now
        consider y9 being Element of the AffinStruct of POS such that
A17:    a9,b9 // q9,y9 & q9<>y9 by DIRAF:40;
        assume that
A18:    x<>q and
A19:    x<>s;
        not q9,y9 // x9,s9
        proof
          assume not thesis;
          then q9,y9 // r9,s9 by A13,A19,AFF_1:5;
          then r9,s9 // a9,b9 by A17,AFF_1:5;
          then r,s // a,b by Th36;
          then a,b _|_ a,b by A12,A6,Def7;
          hence contradiction by A7,Def7;
        end;
        then consider z9 being Element of the AffinStruct of POS such that
A20:    LIN q9,y9,z9 and
A21:    LIN x9,s9,z9 by A1,AFF_1:60;
        reconsider z=z9 as Element of POS;
        q9,y9 // q9,z9 by A20,AFF_1:def 1;
        then a9,b9 // q9,z9 by A17,AFF_1:5;
        then
A22:    a,b // q,z by Th36;
A23:    x9,s9 // x9,z9 by A21,AFF_1:def 1;
        then x,s // x,z by Th36;
        then a,b _|_ x,z by A14,A19,Def7;
        then a,b _|_ q,z by A15,Def7;
        then q,z _|_ q,z by A7,A22,Def7;
        then x9,s9 // x9,q9 by A23,Def7;
        then
A24:    LIN x9,s9,q9 by AFF_1:def 1;
        LIN x9,s9,x9 & LIN x9,q9,p9 by A10,AFF_1:6,7;
        then LIN x9,s9,p9 by A18,A24,AFF_1:11;
        then x9,s9 // p9,q9 by A24,AFF_1:10;
        then p9,q9 // r9,s9 by A13,A19,AFF_1:5;
        hence contradiction by A7,Th36;
      end;
      r9,s9 // r9,x9 by A11,AFF_1:def 1;
      then
A25:  r9,s9 // x9,r9 by AFF_1:4;
      then r,s // x,r by Th36;
      then a,b _|_ x,r by A12,A6,Def7;
      then
A26:  x,r _|_ a,b by Def7;
A27:  now
        consider y9 being Element of the AffinStruct of POS such that
A28:    a9,b9 // q9,y9 & q9<>y9 by DIRAF:40;
        assume that
A29:    x<>q and
A30:    x<>r;
        not q9,y9 // x9,r9
        proof
          assume not thesis;
          then q9,y9 // r9,s9 by A25,A30,AFF_1:5;
          then r9,s9 // a9,b9 by A28,AFF_1:5;
          then r,s // a,b by Th36;
          then a,b _|_ a,b by A12,A6,Def7;
          hence contradiction by A7,Def7;
        end;
        then consider z9 being Element of the AffinStruct of POS such that
A31:    LIN q9,y9,z9 and
A32:    LIN x9,r9,z9 by A1,AFF_1:60;
        reconsider z=z9 as Element of POS;
        q9,y9 // q9,z9 by A31,AFF_1:def 1;
        then a9,b9 // q9,z9 by A28,AFF_1:5;
        then
A33:    a,b // q,z by Th36;
A34:    x9,r9 // x9,z9 by A32,AFF_1:def 1;
        then x,r // x,z by Th36;
        then a,b _|_ x,z by A26,A30,Def7;
        then a,b _|_ q,z by A15,Def7;
        then q,z _|_ q,z by A7,A33,Def7;
        then x9,r9 // x9,q9 by A34,Def7;
        then
A35:    LIN x9,r9,q9 by AFF_1:def 1;
        LIN x9,r9,x9 & LIN x9,q9,p9 by A10,AFF_1:6,7;
        then LIN x9,r9,p9 by A29,A35,AFF_1:11;
        then x9,r9 // p9,q9 by A35,AFF_1:10;
        then p9,q9 // r9,s9 by A25,A30,AFF_1:5;
        hence contradiction by A7,Th36;
      end;
      p9,q9 // p9,x9 by A10,AFF_1:def 1;
      then p9,q9 // x9,p9 by AFF_1:4;
      then p,q // x,p by Th36;
      then
A36:  a,b _|_ x,p by A9,A5,Def7;
A37:  now
        consider y9 being Element of the AffinStruct of POS such that
A38:    a9,b9 // p9,y9 & p9<>y9 by DIRAF:40;
        assume that
A39:    x<>p and
A40:    x<>s;
        not p9,y9 // x9,s9
        proof
          assume not thesis;
          then p9,y9 // r9,s9 by A13,A40,AFF_1:5;
          then r9,s9 // a9,b9 by A38,AFF_1:5;
          then r,s // a,b by Th36;
          then a,b _|_ a,b by A12,A6,Def7;
          hence contradiction by A7,Def7;
        end;
        then consider z9 being Element of the AffinStruct of POS such that
A41:    LIN p9,y9,z9 and
A42:    LIN x9,s9,z9 by A1,AFF_1:60;
        reconsider z=z9 as Element of POS;
        p9,y9 // p9,z9 by A41,AFF_1:def 1;
        then a9,b9 // p9,z9 by A38,AFF_1:5;
        then
A43:    a,b // p,z by Th36;
A44:    x9,s9 // x9,z9 by A42,AFF_1:def 1;
        then x,s // x,z by Th36;
        then a,b _|_ x,z by A14,A40,Def7;
        then a,b _|_ p,z by A36,Def7;
        then p,z _|_ p,z by A7,A43,Def7;
        then x9,s9 // x9,p9 by A44,Def7;
        then
A45:    LIN x9,s9,p9 by AFF_1:def 1;
        LIN x9,s9,x9 & LIN x9,p9,q9 by A10,AFF_1:6,7;
        then LIN x9,s9,q9 by A39,A45,AFF_1:11;
        then x9,s9 // p9,q9 by A45,AFF_1:10;
        then p9,q9 // r9,s9 by A13,A40,AFF_1:5;
        hence contradiction by A7,Th36;
      end;
      now
        consider y9 being Element of the AffinStruct of POS such that
A46:    a9,b9 // p9,y9 & p9<>y9 by DIRAF:40;
        assume that
A47:    x<>p and
A48:    x<>r;
        not p9,y9 // x9,r9
        proof
          assume not thesis;
          then p9,y9 // r9,s9 by A25,A48,AFF_1:5;
          then r9,s9 // a9,b9 by A46,AFF_1:5;
          then r,s // a,b by Th36;
          then a,b _|_ a,b by A12,A6,Def7;
          hence contradiction by A7,Def7;
        end;
        then consider z9 being Element of the AffinStruct of POS such that
A49:    LIN p9,y9,z9 and
A50:    LIN x9,r9,z9 by A1,AFF_1:60;
        reconsider z=z9 as Element of POS;
        p9,y9 // p9,z9 by A49,AFF_1:def 1;
        then a9,b9 // p9,z9 by A46,AFF_1:5;
        then
A51:    a,b // p,z by Th36;
A52:    x9,r9 // x9,z9 by A50,AFF_1:def 1;
        then x,r // x,z by Th36;
        then a,b _|_ x,z by A26,A48,Def7;
        then a,b _|_ p,z by A36,Def7;
        then p,z _|_ p,z by A7,A51,Def7;
        then x9,r9 // x9,p9 by A52,Def7;
        then
A53:    LIN x9,r9,p9 by AFF_1:def 1;
        LIN x9,r9,x9 & LIN x9,p9,q9 by A10,AFF_1:6,7;
        then LIN x9,r9,q9 by A47,A53,AFF_1:11;
        then x9,r9 // p9,q9 by A53,AFF_1:10;
        then p9,q9 // r9,s9 by A25,A48,AFF_1:5;
        hence contradiction by A7,Th36;
      end;
      hence contradiction by A8,A37,A27,A16,AFF_1:3;
    end;
  end;
  for a,b,c being Element of POS ex x being Element of POS st a,b _|_ c,x
  & c <>x by Def7;
  hence thesis by A1,A2,Def8;
end;
