theorem Th45:
  K _|_ M iff ex a,b,c,d st a<>b & c <>d & K = Line(a,b) & M =
  Line(c,d) & a,b _|_ c,d
proof
  hereby
    assume K _|_ M;
    then consider a,b such that
A1: a<>b & K = Line(a,b) and
A2: a,b _|_ M;
    ex c,d st c <>d & M = Line(c,d) & a,b _|_ c,d by A2;
    hence ex a,b,c,d st a<>b & c <>d & K = Line(a,b) & M = Line(c,d) & a,b _|_
    c,d by A1;
  end;
  given a,b,c,d such that
A3: a<>b and
A4: c <>d and
A5: K = Line(a,b) and
A6: M = Line(c,d) & a,b _|_ c,d;
  a,b _|_ M by A4,A6;
  hence thesis by A3,A5;
end;
