reserve x,y for set;
reserve X for non empty set;
reserve a,b,c,d for Element of X;
reserve S for OAffinSpace;
reserve a,b,c,d,p,q,r,x,y,z,t,u,w for Element of S;
reserve AS for non empty AffinStruct;
reserve S for OAffinPlane;
reserve x,y,z,t,u for Element of S;

theorem Th44:
  AS = Lambda(S) implies for x,y,z,t being Element of AS st not x,
  y // z,t ex u being Element of AS st x,y // x,u & z,t // z,u
proof
  assume
A1: AS = Lambda(S);
  let x,y,z,t be Element of AS;
  reconsider x9=x, y9=y, z9=z, t9=t as Element of S by A1;
  assume not x,y // z,t;
  then not x9,y9 '||' z9,t9 by A1,Th38;
  then consider u9 being Element of S such that
A2: x9,y9 '||' x9,u9 & z9,t9 '||' z9,u9 by Th43;
  reconsider u=u9 as Element of AS by A1;
  x,y // x,u & z,t // z,u by A1,A2,Th38;
  hence thesis;
end;
