
theorem Th39:
  for a,b,c,d,e,f,r being Real st
  (1 - r) * |[a,b,1]| + r * |[c,d,1]| = |[e,f,1]| holds
  (1 - r) * |[a,b]| + r * |[c,d]| = |[e,f]|
  proof
    let a,b,c,d,e,f,r be Real;
    assume (1 - r) * |[a,b,1]| + r * |[c,d,1]| = |[e,f,1]|;
    then |[e,f,1]|=|[(1 - r) * a,(1 - r) * b,(1 - r) * 1]| + r * |[c,d,1]|
      by EUCLID_5:8
                 .=|[(1 - r) * a,(1 - r) * b,(1 - r)]|
                     + |[r * c,r * d,r * 1]| by EUCLID_5:8
                 .=|[(1 - r) * a + r * c,(1 - r) * b + r * d,1 - r + r]|
                     by EUCLID_5:6;
    then e = (1 - r) * a + r * c & f = (1 - r) * b + r * d by FINSEQ_1:78;
    then |[e,f]| = |[ (1 - r) * a, (1 - r) * b ]| + |[ r * c,r * d]|
      by EUCLID:56
                .= (1 - r) * |[a,b]| + |[r * c,r * d]| by EUCLID:58
                .= (1 - r) * |[a,b]| + r * |[c,d]| by EUCLID:58;
    hence thesis;
  end;
