reserve V for RealLinearSpace,
  o,p,q,r,s,u,v,w,y,y1,u1,v1,w1,u2,v2,w2 for Element of V,
  a,b,c,d,a1,b1,c1,d1,a2,b2,c2,d2,a3,b3,c3,d3 for Real,
  z for set;
reserve A for non empty set;
reserve f,g,h,f1 for Element of Funcs(A,REAL);
reserve x1,x2,x3,x4 for Element of A;
reserve V for non trivial RealLinearSpace;
reserve u,v,w,y,u1,v1,w1,u2,w2 for Element of V;
reserve p,p1,p2,p3,q,q1,q2,q3,r,r1,r2,r3 for Element of ProjectiveSpace(V);

theorem Th25:
  (ex u,v st for a,b st a*u + b*v = 0.V holds a = 0 & b = 0 )
  implies ProjectiveSpace(V) is at_least_3rank
proof
  given u,v such that
A1: for a,b st a*u + b*v = 0.V holds a = 0 & b = 0;
A2: not are_Prop u,v by A1,Lm1;
  let p,q;
  consider y such that
A3: y is not zero & p = Dir(y) by ANPROJ_1:26;
  consider w such that
A4: w is not zero & q = Dir(w) by ANPROJ_1:26;
  u is not zero & v is not zero by A1,Lm1;
  then consider z being Element of V such that
A5: z is not zero and
A6: y,w,z are_LinDep and
A7: not are_Prop y,z and
A8: not are_Prop w,z by A2,ANPROJ_1:16;
  reconsider r = Dir(z) as Element of ProjectiveSpace(V) by A5,ANPROJ_1:26;
  take r;
  thus p<>r by A3,A5,A7,ANPROJ_1:22;
  thus q<>r by A4,A5,A8,ANPROJ_1:22;
  thus thesis by A3,A4,A5,A6,Th23;
end;
