reserve a,b,c,d,e,f for Real,
        g           for positive Real,
        x,y         for Complex,
        S,T         for Element of REAL 2,
        u,v,w       for Element of TOP-REAL 3;

theorem
  a^2 + b^2 < 1 & c^2 + d^2 = 1 implies ((a + c)/2)^2 + ((b + d)/2)^2 < 1
  proof
    assume that
A1: a^2 + b^2 < 1 and
A2: c^2 + d^2 = 1;
    reconsider u = |[a,b]|, v = |[c,d]| as Element of TOP-REAL 2;
A4: |. |( u,v )| .| <= |.u.| * |.v.| by EUCLID_2:51;
A5: |.u.|^2 < 1 & |.v.|^2 = 1 by A1,A2,TOPGEN_5:9;
A6: |(u,v)| = u`1 * v`1 + u`2 * v`2 by EUCLID_3:41
           .= a * v`1 + u`2 * v`2 by EUCLID:52
           .= a * v`1 + b * v`2 by EUCLID:52
           .= a * c + b * v`2 by EUCLID:52
           .= a * c + b * d by EUCLID:52;
           |.u.|^2 * |.v.|^2 < 1 * 1 by A5;
    then (|.u.| * |.v.|)^2 < 1;
    then |.u.| * |.v.| < 1 by SQUARE_1:52; then
A7: |. |(u,v)| .| < 1 by XXREAL_0:2,A4;
    |(u,v)| <= |. |(u,v)| .| by COMPLEX1:76;
    then a * c + b * d < 1 by A6,A7,XXREAL_0:2;
    then 2 * (a * c + b * d) < 2 * 1 by XREAL_1:68;
    then (2 * a * c + 2 * b * d) + (a^2 + b^2) < 2 + 1 by A1,XREAL_1:8;
    then (2 * a * c + 2 * b * d) + (a^2 + b^2) + 1 < 2 + 1 + 1 by XREAL_1:8;
    then ((2 * a * c + 2 * b * d) + (a^2 + b^2) + 1)/4 < 4/4 by XREAL_1:74;
    hence thesis by A2;
  end;
