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 Th1:
  (for a,b,c st a*u + b*v + c*w = 0.V holds a = 0 & b = 0 & c = 0)
implies u is not zero & v is not zero & w is not zero & not u,v,w are_LinDep &
  not are_Prop u,v
proof
  assume
A1: for a,b,c st a*u + b*v + c*w = 0.V holds a=0 & b=0 & c = 0;
A2: now
    assume not v is not zero;
    then
A3: v = 0.V;
    0.V = 0.V + 0.V by RLVECT_1:4
      .= 0.V + 0.V + 0.V by RLVECT_1:4
      .= 0.V + 1*v + 0.V by A3,RLVECT_1:def 8
      .= 0*u + 1*v + 0.V by RLVECT_1:10
      .= 0*u + 1*v + 0*w by RLVECT_1:10;
    hence contradiction by A1;
  end;
A4: now
    assume not w is not zero;
    then
A5: w = 0.V;
    0.V = 0.V + 0.V by RLVECT_1:4
      .= 0.V + 0.V + 0.V by RLVECT_1:4
      .= 0.V + 0.V + 1*w by A5,RLVECT_1:def 8
      .= 0*u + 0.V + 1*w by RLVECT_1:10
      .= 0*u + 0*v + 1*w by RLVECT_1:10;
    hence contradiction by A1;
  end;
  now
    assume not u is not zero;
    then
A6: u = 0.V;
    0.V = 0.V + 0.V by RLVECT_1:4
      .= 0.V + 0.V + 0.V by RLVECT_1:4
      .= 1*u + 0.V + 0.V by A6,RLVECT_1:def 8
      .= 1*u + 0*v + 0.V by RLVECT_1:10
      .= 1*u + 0*v + 0*w by RLVECT_1:10;
    hence contradiction by A1;
  end;
  hence u is not zero & v is not zero & w is not zero by A2,A4;
  thus not u,v,w are_LinDep by A1;
  hence thesis by ANPROJ_1:12;
end;
