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;
reserve POS for OrtAfSp;
reserve A,K,M,N for Subset of POS;
reserve a,b,c,d,p,q,r,s for Element of POS;

theorem Th56:
  a in M & b in M & c in N & d in N & M _|_ N implies a,b _|_ c,d
proof
  assume that
A1: a in M and
A2: b in M and
A3: c in N and
A4: d in N and
A5: M _|_ N;
  consider p1,q1,p2,q2 being Element of POS such that
A6: p1<>q1 and
A7: p2<>q2 and
A8: M = Line(p1,q1) and
A9: N = Line(p2,q2) and
A10: p1,q1 _|_ p2,q2 by A5,Th45;
  reconsider a9=a,b9=b,c9=c,d9=d,p19=p1,q19=q1,p29=p2,q29=q2
    as Element of the AffinStruct of POS;
  LIN p1,q1,b by A2,A8,Def10;
  then
A11: LIN p19,q19,b9 by Th40;
  LIN p1,q1,a by A1,A8,Def10;
  then LIN p19,q19,a9 by Th40;
  then p19,q19 // a9,b9 by A11,AFF_1:10;
  then p1,q1 // a,b by Th36;
  then
A12: p2,q2 _|_ a,b by A6,A10,Def7;
  LIN p2,q2,d by A4,A9,Def10;
  then
A13: LIN p29,q29,d9 by Th40;
  LIN p2,q2,c by A3,A9,Def10;
  then LIN p29,q29,c9 by Th40;
  then p29,q29 // c9,d9 by A13,AFF_1:10;
  then p2,q2 // c,d by Th36;
  hence thesis by A7,A12,Def7;
end;
