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;

theorem
  x,y,z are_collinear & x,y,t are_collinear implies x,y '||' z,t
proof
  assume
A1: x,y,z are_collinear & x,y,t are_collinear;
  now
A2: x,y '||' x,z & x,y '||' x,t by A1;
    assume x<>y;
    then x,z '||' x,t by A2,Th23;
    then z,x '||' z,t by Th21;
    then x,z '||' z,t by Th22;
    hence thesis by A2,Th23;
  end;
  hence thesis by Th20;
end;
