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
  a,b _|_ K implies ex p st LIN a,b,p & p in K
proof
  assume a,b _|_ K;
  then consider p,q such that
  p<>q and
A1: K = Line(p,q) and
A2: a,b _|_ p,q;
  consider c such that
A3: LIN a,b,c and
A4: LIN p,q,c by A2,Th67;
  c in K by A1,A4,Def10;
  hence thesis by A3;
end;
