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 Th67:
  a,b _|_ c,d implies ex p st LIN a,b,p & LIN c,d,p
proof
  reconsider a9=a,b9=b,c9=c,d9=d as Element of the AffinStruct of POS;
  assume
A1: a,b _|_ c,d;
A2: now
    set M = Line(a,b),N = Line(c,d);
    assume a<>b & c <>d;
    then M _|_ N by A1,Th45;
    then consider p such that
A3: p in M & p in N by Th66;
    LIN a,b,p & LIN c,d,p by A3,Def10;
    hence thesis;
  end;
  LIN a9,b9,a9 by AFF_1:7;
  then
A4: LIN a,b,a by Th40;
A5: now
    assume c =d;
    then c,d // c,a by Th58;
    then LIN c,d,a;
    hence thesis by A4;
  end;
  LIN c9,d9,c9 by AFF_1:7;
  then
A6: LIN c,d,c by Th40;
  now
    assume a=b;
    then a,b // a,c by Th58;
    then LIN a,b,c;
    hence thesis by A6;
  end;
  hence thesis by A5,A2;
end;
