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 Th69:
  ex x st a,x _|_ p,q & LIN p,q,x
proof
A1: now
    assume p<>q;
    then consider x such that
A2: p,q // p,x & p,q _|_ x,a by Def7;
    take x;
    thus a,x _|_ p,q & LIN p,q,x by A2,Th61;
  end;
  now
    assume
A3: p=q;
    take x=a;
    p,q // p,a by A3,Th58;
    hence a,x _|_ p,q & LIN p,q,x by Th58;
  end;
  hence thesis by A1;
end;
