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 Th24:
  (ex p,q being VECTOR of V st (for w being VECTOR of V ex a,b
being Real st a*p + b*q=w)) implies
 for a,b,c,d being Element of OASpace(V) st
not a,b // c,d & not a,b // d,c ex t being Element of OASpace(V) st (a,b // a,t
  or a,b // t,a) & (c,d // c,t or c,d // t,c)
proof
  assume
A1: ex p,q being VECTOR of V st for w being VECTOR of V ex a,b being
  Real st a*p + b*q=w;
  set S = OASpace(V);
  let a,b,c,d be Element of OASpace(V);
  reconsider a9=a,b9=b,c9 = c,d9=d as Element of V;
  assume
  ( not [[a,b],[c,d]] in the CONGR of S)& not [[a,b],[d,c]] in the CONGR of S;
  then ( not a9,b9 // c9,d9)& not a9,b9 // d9,c9 by Th22;
  then consider t9 being VECTOR of V such that
A2: a9,b9 // a9,t9 or a9,b9 // t9,a9 and
A3: c9,d9 // c9,t9 or c9,d9 // t9,c9 by A1,Th21;
  reconsider t=t9 as Element of S;
  [[c,d],[c,t]] in the CONGR of S or [[c,d],[t,c]] in the CONGR of S by A3,Th22
;
  then
A4: c,d // c,t or c,d // t,c;
  [[a,b],[a,t]] in the CONGR of S or [[a,b],[t,a]] in the CONGR of S by A2,Th22
;
  then a,b // a,t or a,b // t,a;
  hence thesis by A4;
end;
