reserve a,b,c,d for Real;
reserve z,z1,z2 for Complex;

theorem Th56:
  |.z1 + z2.| <= |.z1.| + |.z2.|
proof
  set r1 = Re z1, r2 = Re z2, i1 = Im z1, i2 = Im z2;
A1: (Im(z1 + z2))^2 = (i1 + i2)^2 by Th8
    .= i1^2 + 2*i1*i2 + i2^2;
A2: 0 <= r1^2+i1^2 by Lm1;
  (r1^2+i1^2)*(r2^2+i2^2)-(r1*r2+i1*i2)^2 = (r1*i2-i1*r2)^2;
  then 0 <= (r1^2+i1^2)*(r2^2+i2^2)-(r1*r2+i1*i2)^2 by XREAL_1:63;
  then
A3: (r1*r2+i1*i2)^2+0 <= (r1^2+i1^2)*(r2^2+i2^2) by XREAL_1:19;
  r1*r2+i1*i2 <= |.r1*r2+i1*i2.| by Lm29;
  then
A4: r1*r2+i1*i2 <= sqrt (r1*r2+i1*i2)^2 by Lm28;
A5: 0 <= r2^2+i2^2 by Lm1;
  then
A6: (sqrt (r2^2+i2^2))^2 = r2^2+i2^2 by SQUARE_1:def 2;
  0<=(r1*r2+i1*i2)^2 by XREAL_1:63;
  then sqrt (r1*r2+i1*i2)^2 <= sqrt ((r1^2+i1^2)*(r2^2+i2^2)) by A3,SQUARE_1:26
;
  then sqrt (r1*r2+i1*i2)^2 <= sqrt (r1^2+i1^2)*sqrt (r2^2+i2^2) by A2,A5,
SQUARE_1:29;
  then
A7: r1*r2 + i1*i2 <= sqrt (r1^2+i1^2)*sqrt (r2^2+i2^2) by A4,XXREAL_0:2;
  (2*r1*r2 + 2*i1*i2) = 2*(r1*r2 + i1*i2);
  then (2*r1*r2 + 2*i1*i2) <= 2*(sqrt (r1^2+i1^2)*sqrt (r2^2+i2^2)) by A7,
XREAL_1:64;
  then
A8: (r1^2 + i1^2)+(2*r1*r2+2*i1*i2) <= (r1^2+i1^2)+2*sqrt (r1^2+i1^2)*sqrt
  (r2^2+i2^2) by XREAL_1:7;
  (Re(z1 + z2))^2 = (r1+ r2)^2 by Th8
    .= r1^2 + 2*r1*r2 + r2^2;
  then
  (Re(z1+z2))^2+(Im(z1+z2))^2 = r1^2 + i1^2 + (2*r1*r2 + 2*i1*i2) + (r2^2
  + i2^2) by A1;
  then
A9: (Re(z1+z2))^2+(Im(z1+z2))^2 <= (r1^2+i1^2)+2*sqrt (r1^2+i1^2)*sqrt (r2
  ^2+i2^2)+(r2^2+i2^2) by A8,XREAL_1:7;
A10: 0 <= (Re(z1 + z2))^2 + (Im(z1 + z2))^2 by Lm1;
  (sqrt (r1^2+i1^2))^2 = r1^2+i1^2 by A2,SQUARE_1:def 2;
  then sqrt ((Re(z1 + z2))^2 + (Im(z1 + z2))^2) <= sqrt ((sqrt (r1^2+i1^2) +
  sqrt (r2^2+i2^2))^2) by A6,A9,A10,SQUARE_1:26;
  hence thesis by SQUARE_1:22;
end;
