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
  p<>q & ( p,q // a,b & p,q // c,d or p,q // a,b & c,d // p,q or a,b //
  p,q & c,d // p,q or a,b // p,q & p,q // c,d ) implies a,b // c,d
proof
  assume that
A1: p<>q and
A2: p,q // a,b & p,q // c,d or p,q // a,b & c,d // p,q or a,b // p,q & c
  ,d // p,q or a,b // p,q & p,q // c,d;
  reconsider p9=p,q9=q,a9=a, b9=b,c9= c,d9=d
    as Element of the AffinStruct of POS;
  p9,q9 // a9,b9 & p9,q9 // c9,d9 or p9,q9 // a9,b9 & c9,d9 // p9,q9 or a9
  ,b9 // p9,q9 & c9,d9 // p9,q9 or a9,b9 // p9,q9 & p9,q9 // c9,d9 by A2,Th36;
  then a9,b9 // c9,d9 by A1,AFF_1:5;
  hence thesis by Th36;
end;
