reserve V for RealLinearSpace;
reserve p,q,u,v,w,y for VECTOR of V;
reserve a,b,c,d for Real;
reserve AS for non empty AffinStruct;
reserve a,b,c,d for Element of AS;
reserve x,z for object;

theorem Th23:
  (ex u,v st for a,b being Real st a*u + b*v = 0.V holds a=0 & b=0
  ) implies (ex a,b being Element of OASpace(V) st a<>b) & (for a,b,c,d,p,q,r,s
being Element of OASpace(V) holds a,b // c,c & (a,b // b,a implies a=b) & (a<>b
& a,b // p,q & a,b // r,s implies p,q // r,s) & (a,b // c,d implies b,a // d,c)
& (a,b // b,c implies a,b // a,c) & (a,b // a,c implies a,b // b,c or a,c // c,
b)) & (ex a,b,c,d being Element of OASpace(V) st not a,b // c,d & not a,b // d,
c) & (for a,b,c being Element of OASpace(V) ex d being Element of OASpace(V) st
a,b // c,d & a,c // b,d & b<>d) & for p,a,b,c being Element of OASpace(V) st p
  <>b & b,p // p,c ex d being Element of OASpace(V) st a,p // p,d & a,b // c,d
proof
  given u,v such that
A1: for a,b being Real st a*u + b*v = 0.V holds a=0 & b=0;
  set S = OASpace(V);
A2: u<>v by A1,Th19;
  hence ex a,b being Element of S st a<>b;
  thus for a,b,c,d,p,q,r,s being Element of S holds a,b // c,c & (a,b // b,a
implies a=b) & (a<>b & a,b // p,q & a,b // r,s implies p,q // r,s) & (a,b // c,
d implies b,a // d,c) & (a,b // b,c implies a,b // a,c) & (a,b // a,c implies a
  ,b // b,c or a,c // c,b)
  proof
    let a,b,c,d,p,q,r,s be Element of S;
    reconsider a9=a,b9=b,c9=c,d9=d,p9=p,q9=q,r9=r,s9=s as Element of V;
    a9,b9 // c9,c9;
    hence [[a,b],[c,c]] in the CONGR of S by Def3;
    thus a,b // b,a implies a=b
    by Th22,Th10;
    thus a<>b & a,b // p,q & a,b // r,s implies p,q // r,s
    proof
      assume that
A3:   a<>b and
A4:   [[a,b],[p,q]] in the CONGR of S & [[a,b],[r,s]] in the CONGR of S;
      a9,b9 // p9,q9 & a9,b9 // r9,s9 by A4,Th22;
      then p9,q9 // r9,s9 by A3,Th11;
      then [[p,q],[r,s]] in the CONGR of S by Th22;
      hence thesis;
    end;
    thus a,b // c,d implies b,a // d,c
    proof
      assume [[a,b],[c,d]] in the CONGR of S;
      then a9,b9 // c9,d9 by Th22;
      then b9,a9 // d9,c9 by Th12;
      then [[b,a],[d,c]] in the CONGR of S by Th22;
      hence thesis;
    end;
    thus a,b // b,c implies a,b // a,c
    proof
      assume [[a,b],[b,c]] in the CONGR of S;
      then a9,b9 // b9,c9 by Th22;
      then a9,b9 // a9,c9 by Th13;
      then [[a,b],[a,c]] in the CONGR of S by Th22;
      hence thesis;
    end;
    thus a,b // a,c implies a,b // b,c or a,c // c,b
    proof
      assume [[a,b],[a,c]] in the CONGR of S;
      then a9,b9 // a9,c9 by Th22;
      then a9,b9 // b9,c9 or a9,c9 // c9,b9 by Th14;
      then [[a,b],[b,c]] in the CONGR of S or [[a,c],[c,b]] in the CONGR of S
      by Th22;
      hence thesis;
    end;
  end;
  thus ex a,b,c,d being Element of S st not a,b // c,d & not a,b // d,c
  proof
    consider a9,b9,c9,d9 being VECTOR of V such that
A5: not a9,b9 // c9,d9 and
A6: not a9,b9 // d9,c9 by A1,Th20;
    reconsider a=a9,b=b9,c = c9,d=d9 as Element of S;
    not [[a,b],[d,c]] in the CONGR of S by A6,Th22;
    then
A7: not a,b // d,c;
    not [[a,b],[c,d]] in the CONGR of S by A5,Th22;
    then not a,b // c,d;
    hence thesis by A7;
  end;
  thus for a,b,c being Element of S ex d being Element of S st a,b // c,d & a,
  c // b,d & b<>d
  proof
    let a,b,c be Element of S;
    reconsider a9=a,b9=b,c9=c as Element of V;
    consider d9 being VECTOR of V such that
A8: a9,b9 // c9,d9 and
A9: a9,c9 // b9,d9 and
A10: b9<>d9 by A2,Th17;
    reconsider d=d9 as Element of S;
    [[a,c],[b,d]] in the CONGR of S by A9,Th22;
    then
A11: a,c // b,d;
    [[a,b],[c,d]] in the CONGR of S by A8,Th22;
    then a,b // c,d;
    hence thesis by A10,A11;
  end;
  thus for p,a,b,c being Element of S st p<>b & b,p // p,c holds ex d being
  Element of S st a,p // p,d & a,b // c,d
  proof
    let p,a,b,c be Element of S;
    assume that
A12: p<>b and
A13: [[b,p],[p,c]] in the CONGR of S;
    reconsider p9=p,a9=a,b9=b,c9=c as Element of V;
    b9,p9 // p9,c9 by A13,Th22;
    then consider d9 being VECTOR of V such that
A14: a9,p9 // p9,d9 and
A15: a9,b9 // c9,d9 by A12,Th18;
    reconsider d=d9 as Element of S;
    [[a,b],[c,d]] in the CONGR of S by A15,Th22;
    then
A16: a,b // c,d;
    [[a,p],[p,d]] in the CONGR of S by A14,Th22;
    then a,p // p,d;
    hence thesis by A16;
  end;
end;
