theorem Th60:
  f is dilatation & f.p=p & Mid q,p,f.q & q<>p implies Mid x,p,f.x
proof
  assume that
A1: f is dilatation & f.p=p and
A2: Mid q,p,f.q and
A3: q<>p;
  now
    consider r such that
A4: not p,q,r are_collinear by A3,DIRAF:37;
    assume
A5: p,q,x are_collinear;
A6: x=p or not p,r,x are_collinear
    proof
A7:   p,x,q are_collinear & p,x,p are_collinear by A5,DIRAF:30,31;
      assume that
A8:   x<>p and
A9:   p,r,x are_collinear;
      p,x,r are_collinear by A9,DIRAF:30;
      hence contradiction by A4,A8,A7,DIRAF:32;
    end;
    Mid r,p,f.r by A1,A2,A4,Th59;
    hence thesis by A1,A6,Th59,DIRAF:10;
  end;
  hence thesis by A1,A2,Th59;
end;
