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 Th50:
  r,s _|_ K & K // M implies r,s _|_ M
proof
  assume that
A1: r,s _|_ K and
A2: K // M;
  consider p,q such that
A3: p<>q and
A4: K = Line(p,q) and
A5: r,s _|_ p,q by A1;
  consider a,b,c,d such that
A6: a<>b and
A7: c <>d and
A8: K = Line(a,b) and
A9: M = Line(c,d) and
A10: a,b // c,d by A2;
  reconsider p9=p,q9=q,a9=a,b9=b,c9=c,d9=d
    as Element of the AffinStruct of POS;
A11: K = Line(a9,b9) by A8,Th41;
A12: K = Line(p9,q9) by A4,Th41;
  then q9 in K by AFF_1:15;
  then
A13: LIN a9,b9,q9 by A11,AFF_1:def 2;
  p9 in K by A12,AFF_1:15;
  then LIN a9,b9,p9 by A11,AFF_1:def 2;
  then
A14: a9,b9 // p9,q9 by A13,AFF_1:10;
A15: p,q _|_ r,s by A5,Def7;
  a9,b9 // c9,d9 by A10,Th36;
  then p9,q9 // c9,d9 by A6,A14,AFF_1:5;
  then p,q // c,d by Th36;
  then r,s _|_ c,d by A3,A15,Def7;
  hence thesis by A7,A9;
end;
