reserve V for RealLinearSpace;
reserve u,u1,u2,v,v1,v2,w,w1,y for VECTOR of V;
reserve a,a1,a2,b,b1,b2,c1,c2 for Real;
reserve x,z for set;
reserve p,p1,q,q1 for Element of Lambda(OASpace(V));
reserve POS for non empty ParOrtStr;
reserve p,p1,p2,q,q1,r,r1,r2 for Element of AMSpace(V,w,y);
reserve x,a,b,c,d,p,q,y for Element of POS;
reserve A,K,M for Subset of POS;

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;
