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;

theorem Th5:
  (for a,b,a1,b1 st a*u + b*v + a1*u1 + b1*v1 = 0.V holds a=0 & b=0
  & a1=0 & b1=0) implies not ex y st y is not zero & u,v,y are_LinDep & u1,v1,y
  are_LinDep
proof
  assume
A1: for a,b,a1,b1 st a*u + b*v + a1*u1 + b1*v1 = 0.V holds a=0 & b=0 &
  a1=0 & b1=0;
  then
A2: not are_Prop u,v by Th2;
  assume not thesis;
  then consider y such that
A3: y is not zero and
A4: u,v,y are_LinDep and
A5: u1,v1,y are_LinDep;
  u is not zero & v is not zero by A1,Th2;
  then consider a,b such that
A6: y = a*u + b*v by A4,A2,ANPROJ_1:6;
A7: not are_Prop u1,v1 by A1,Th2;
  u1 is not zero & v1 is not zero by A1,Th2;
  then consider a1,b1 such that
A8: y = a1*u1 + b1*v1 by A5,A7,ANPROJ_1:6;
  0.V = (a*u + b*v) - (a1*u1 + b1*v1) by A6,A8,RLVECT_1:15
    .= (a*u + b*v) + (-1)*(a1*u1 + b1*v1) by RLVECT_1:16
    .= (a*u + b*v) + ((-1)*(a1*u1) + (-1)*(b1*v1)) by RLVECT_1:def 5
    .= (a*u + b*v) + (((-1)*a1)*u1 + (-1)*(b1*v1)) by RLVECT_1:def 7
    .= (a*u + b*v) + (((-1)*a1)*u1 + ((-1)*b1)*v1) by RLVECT_1:def 7
    .= a*u + b*v + ((-1)*a1)*u1 + ((-1)*b1)*v1 by RLVECT_1:def 3;
  then a=0 & b=0 by A1;
  then y = 0.V + 0*v by A6,RLVECT_1:10
    .= 0.V + 0.V by RLVECT_1:10
    .= 0.V by RLVECT_1:4;
  hence contradiction by A3;
end;
