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 Th32:
  x<>y & x,y,z are_collinear & x,y,t are_collinear & x,y,u
  are_collinear implies z,t,u are_collinear
proof
  assume that
A1: x<>y and
A2: x,y,z are_collinear and
A3: x,y,t are_collinear and
A4: x,y,u are_collinear;
A5: now
A6: x,y '||' x,z by A2;
    x,y '||' x,u by A4;
    then x,z '||' x,u by A1,A6,Th23;
    then
A7: z,x '||' z,u by Th21;
    x,y '||' x,t by A3;
    then x,z '||' x,t by A1,A6,Th23;
    then
A8: z,x '||' z,t by Th21;
    assume x<>z;
    then z,t '||' z,u by A8,A7,Th23;
    hence thesis;
  end;
  x=z implies thesis by A3,A4,A1,Th23;
  hence thesis by A5;
end;
