
theorem Th29:
  for n31,n32,n33 being Element of F_Real
  for u,v being Element of TOP-REAL 2 st u in inside_of_circle(0,0,1) &
  v in inside_of_circle(0,0,1) &
  (for w being Element of TOP-REAL 2 st w in inside_of_circle(0,0,1) holds
  n31 * w.1 + n32 * w.2 + n33 <> 0) holds
  0 < (n31 * u.1 + n32 * u.2 + n33) * (n31 * v.1 + n32 * v.2 + n33)
  proof
    let n31,n32,n33 be Element of F_Real;
    let u,v be Element of TOP-REAL 2;
    assume that
A1: u in inside_of_circle(0,0,1) and
A2: v in inside_of_circle(0,0,1) and
A3: for w be Element of TOP-REAL 2 st w in inside_of_circle(0,0,1) holds
      n31 * w.1 + n32 * w.2 + n33 <> 0;
    set B = inside_of_circle(0,0,1);
    set A = {x where x is Element of TOP-REAL 3:|[x`1,x`2]| in B & x`3 = 1};
    A c= the carrier of TOP-REAL 3
    proof
      let x be object;
      assume x in A;
      then ex y be Element of TOP-REAL 3 st x= y & |[y`1,y`2]| in B & y`3 = 1;
      hence thesis;
    end;
    then reconsider A as Subset of TOP-REAL 3;
    reconsider A as non empty convex Subset of TOP-REAL 3 by Th26;
    reconsider n = |[ n31,n32,n33 ]|, u9 = |[u.1,u.2,1]|,v9 = |[v.1,v.2,1]|
      as Element of TOP-REAL 3;
A4: |[u`1,u`2]| in B & |[v`1,v`2]| in B by A1,A2,EUCLID:53;
A5: u9`1 = u.1 & u9`2 = u.2 & u9`3 = 1 &
    v9`1 = v.1 & v9`2 = v.2 & v9`3 = 1 by EUCLID_5:2;
A6: u9 in A & v9 in A by A5,A4;
A7:
    now
      let w be Element of TOP-REAL 3;
      assume w in A;
      then consider x be Element of TOP-REAL 3 such that
A8:   w = x and
A9:   |[x`1,x`2]| in B and
A10:  x`3 = 1;
      reconsider v = |[x`1,x`2]| as Element of TOP-REAL 2;
      now
        w.3 = 1 by A8,A10,EUCLID_5:def 3;
        hence |(n,w)| = n31 * w.1 + n32 * w.2 + n33 by Th27;
        thus w.1 = w`1 by EUCLID_5:def 1
                .= v`1 by A8,EUCLID:52
                .= v.1;
        thus w.2 = w`2 by EUCLID_5:def 2
                .= v`2 by A8,EUCLID:52
                .= v.2;
      end;
      hence |(n,w)| <> 0 by A3,A9;
    end;
    now
      n = <*n31,n32,n33*> & u9.3 = 1 by A5,EUCLID_5:def 3;
      then |(n,u9)| = n31 * u9.1 + n32 * u9.2 + n33 by Th27;
      then |(n,u9)| = n31 * u9`1 + n32 * u9.2 + n33 by EUCLID_5:def 1;
      hence |(n,u9)| = n31 * u.1 + n32 * u.2 + n33 by A5,EUCLID_5:def 2;
      n = <*n31,n32,n33*> & v9.3 = 1 by A5,EUCLID_5:def 3;
      then |(n,v9)| = n31 * v9.1 + n32 * v9.2 + n33 by Th27;
      then |(n,v9)| = n31 * v9`1 + n32 * v9.2 + n33 by EUCLID_5:def 1;
      hence |(n,v9)| = n31 * v.1 + n32 * v.2 + n33 by A5,EUCLID_5:def 2;
    end;
    hence thesis by A7,A6,Th28;
  end;
