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
  a,b _|_ K & (a,b // c,d or c,d // a,b) & a<>b implies c,d _|_ K
proof
  assume that
A1: a,b _|_ K and
A2: a,b // c,d or c,d // a,b and
A3: a<>b;
  reconsider a9=a,b9=b,c9=c,d9=d as Element of the AffinStruct of POS;
  consider p,q such that
A4: p<>q & K = Line(p,q) and
A5: a,b _|_ p,q by A1;
  a9,b9 // c9,d9 or c9,d9 // a9,b9 by A2,Th36;
  then a9,b9 // c9,d9 by AFF_1:4;
  then a,b // c,d by Th36;
  then p,q _|_ c,d by A3,A5,Def7;
  then c,d _|_ p,q by Def7;
  hence thesis by A4;
end;
