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 Th52:
  K _|_ M & K // N implies N _|_ M
proof
  assume that
A1: K _|_ M and
A2: K // N;
  consider r,s such that
A3: r<>s & M = Line(r,s) and
A4: r,s _|_ K by A1,Def13;
  r,s _|_ N by A2,A4,Th50;
  hence thesis by A3,Def13;
end;
