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 Th37:
  for POS being OrtAfPl holds POS is OrtAfSp
proof
  let POS be OrtAfPl;
  for a,b,c,d,p,q,r,s being Element of POS holds (a,b _|_ p,q & a,b _|_ p,
  s implies a,b _|_ q,s)
  proof
    let a,b,c,d,p,q,r,s be Element of POS such that
A1: a,b _|_ p,q and
A2: a,b _|_ p,s;
A3: now
      reconsider p9=p,q9=q,s9=s as Element of the AffinStruct of POS;
      assume that
A4:   a<>b and
A5:   p<>q;
      p,q // p,s by A1,A2,A4,Def8;
      then p9,q9 // p9,s9 by Th36;
      then q9,p9 // q9,s9 by DIRAF:40;
      then p9,q9 // q9,s9 by AFF_1:4;
      then
A6:   p,q // q,s by Th36;
      p,q _|_ a,b by A1,Def8;
      hence thesis by A5,A6,Def8;
    end;
    now
      assume a=b;
      then q,s _|_ a,b by Def8;
      hence thesis by Def8;
    end;
    hence thesis by A2,A3;
  end;
  then
A7: for a,b,c,d,p,q,r,s be Element of POS holds (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) &( a,b _|_ p,q & a,b _|_ p,s implies a,b
  _|_ q,s) by Def8;
A8: for a,b,c being Element of POS st a<>b ex x being Element of POS st a,b
  // a,x & a,b _|_ x,c
  proof
    let a,b,c be Element of POS such that
A9: a<>b;
    consider y being Element of POS such that
A10: a,b _|_ c,y and
A11: c <>y by Def8;
    reconsider a9=a,b9=b,c9=c,y9=y as Element of the AffinStruct of POS;
    not a9,b9 // c9,y9
    proof
      assume not thesis;
      then a,b // c,y by Th36;
      then c,y _|_ c,y by A9,A10,Def8;
      hence contradiction by A11,Def8;
    end;
    then consider x9 being Element of the AffinStruct of POS such that
A12: LIN a9,b9,x9 and
A13: LIN c9,y9,x9 by AFF_1:60;
    reconsider x=x9 as Element of POS;
    c9,y9 // c9,x9 by A13,AFF_1:def 1;
    then
A14: c,y // c,x by Th36;
    c,y _|_ a,b by A10,Def8;
    then a,b _|_ c,x by A11,A14,Def8;
    then
A15: a,b _|_ x,c by Def8;
    a9,b9 // a9,x9 by A12,AFF_1:def 1;
    then a,b // a,x by Th36;
    hence thesis by A15;
  end;
  the AffinStruct of POS = AffinStruct(#the carrier of POS, the CONGR of POS#)
  & for a,b,c being Element of POS ex x being Element of POS st a,b _|_ c,x
  & c <>x by Def8;
  hence thesis by A8,A7,Def7;
end;
