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 & x,y,z are_collinear & x,y '||' z,t implies x,y,t are_collinear
proof
  assume that
A1: x<>y and
A2: x,y,z are_collinear and
A3: x,y '||' z,t;
  now
    x,y '||' x,z by A2;
    then x,z '||' z,t by A1,A3,Th23;
    then z,x '||' z,t by Th22;
    then z,x,t are_collinear;
    then
A4: x,z,t are_collinear by Th30;
    assume
A5: z<>x;
    x,z,y are_collinear & x,z,x are_collinear by A2,Th30,Th31;
    hence thesis by A5,A4,Th32;
  end;
  hence thesis by A3;
end;
