reserve V for RealLinearSpace;
reserve p,q,r,u,v,w,y,u1,v1,w1 for Element of V;
reserve a,b,c,d,a1,b1,c1,a2,b2,c2,a3,b3,e,f for Real;

theorem
  not are_Prop p,q & p is not zero & q is not zero implies for u,v ex y
  st y is not zero & u,v,y are_LinDep & not are_Prop u,y & not are_Prop v,y
proof
  assume that
A1: not are_Prop p,q and
A2: p is not zero and
A3: q is not zero;
  let u,v;
A4: now
    assume that
    not are_Prop u,v and
A5: not u is not zero;
A6: u=0.V by A5;
    then
A7: not are_Prop v,q implies not are_Prop v,q & q is not zero & u,v,q
    are_LinDep & not are_Prop u,q by A3,Th3,Th10;
    not are_Prop v,p implies not are_Prop v,p & p is not zero & u,v,p
    are_LinDep & not are_Prop u,p by A2,A6,Th3,Th10;
    hence thesis by A1,A7,Th2;
  end;
A8: now
    set y=u+v;
    assume that
A9: not are_Prop u,v and
A10: u is not zero and
A11: v is not zero;
    u+v<>0.V by A9,Th13;
    hence y is not zero;
    1*u+1*v+(-1)*y = u+1*v+(-1)*(u+v) by RLVECT_1:def 8
      .= u+v+(-1)*(u+v) by RLVECT_1:def 8
      .= u + v + -(u+v) by RLVECT_1:16
      .= v+u+(-u+-v) by RLVECT_1:31
      .= v+(u+(-u+-v)) by RLVECT_1:def 3
      .= v+((u+-u)+-v) by RLVECT_1:def 3
      .= v+(0.V+-v) by RLVECT_1:5
      .= v+(-v)
      .= 0.V by RLVECT_1:5;
    hence u,v,y are_LinDep;
A12: v<>0.V by A11;
    now
      let a,b;
      assume a*u = b*y;
      then -b*u + a*u = -b*u + (b*u + b*v) by RLVECT_1:def 5
        .= (b*u + -b*u) + b*v by RLVECT_1:def 3
        .= 0.V + b*v by RLVECT_1:5
        .= b*v;
      then
A13:  b*v = a*u + b*(-u) by RLVECT_1:25
        .= a*u + (-b)*u by RLVECT_1:24
        .= (a + -b)*u by RLVECT_1:def 6;
      now
        assume a + -b = 0;
        then b*v = 0.V by A13,RLVECT_1:10;
        hence b = 0 by A12,RLVECT_1:11;
      end;
      hence a=0 or b=0 by A9,A13;
    end;
    hence not are_Prop u,y;
A14: u<>0.V by A10;
    now
      let a,b;
      assume a*v = b*y;
      then a*v + -b*v = b*u + b*v + -b*v by RLVECT_1:def 5
        .= b*u + (b*v + -b*v) by RLVECT_1:def 3
        .= b*u + 0.V by RLVECT_1:5
        .= b*u;
      then
A15:  b*u = a*v + b*(-v) by RLVECT_1:25
        .= a*v + (-b)*v by RLVECT_1:24
        .= (a + -b)*v by RLVECT_1:def 6;
      now
        assume a + -b = 0;
        then b*u = 0.V by A15,RLVECT_1:10;
        hence b = 0 by A14,RLVECT_1:11;
      end;
      hence a=0 or b=0 by A9,A15;
    end;
    hence not are_Prop v,y;
  end;
A16: now
    assume that
    not are_Prop u,v and
A17: not v is not zero;
A18: v = 0.V by A17;
    then
A19: not are_Prop u,q implies q is not zero & u,v,q are_LinDep & not
    are_Prop u,q & not are_Prop v,q by A3,Th3,Th10;
    not are_Prop u,p implies p is not zero & u,v,p are_LinDep & not
    are_Prop u,p & not are_Prop v,p by A2,A18,Th3,Th10;
    hence thesis by A1,A19,Th2;
  end;
  now
    assume
A20: are_Prop u,v;
    then
A21: not are_Prop u,q implies q is not zero & u,v,q are_LinDep & not
    are_Prop u,q & not are_Prop v,q by A3,Th2,Th11;
    not are_Prop u,p implies p is not zero & u,v,p are_LinDep & not
    are_Prop u,p & not are_Prop v,p by A2,A20,Th2,Th11;
    hence thesis by A1,A21,Th2;
  end;
  hence thesis by A8,A4,A16;
end;
