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;
reserve POS for OrtAfPl;
reserve K,M,N for Subset of POS;
reserve x,a,b,c,d,p,q for Element of POS;

theorem Th66:
  M _|_ N implies ex p st p in M & p in N
proof
  reconsider M9=M,N9=N as Subset of the AffinStruct of POS;
  assume
A1: M _|_ N;
  then M is being_line;
  then
A2: M9 is being_line by Th43;
  N is being_line by A1,Th44;
  then
A3: N9 is being_line by Th43;
  not M // N by A1,Th52;
  then not M9 // N9 by Th46;
  hence thesis by A2,A3,AFF_1:58;
end;
