reserve X for non empty CUNITSTR;
reserve a, b for Complex;
reserve x, y for Point of X;
reserve X for ComplexUnitarySpace;
reserve x, y, z, u, v for Point of X;

theorem Th41:
  ||.x + y.|| <= ||.x.|| + ||.y.||
proof
A1: ||.x + y.||^2 >= 0 by XREAL_1:63;
  Re ((x+y).|.(x+y)) >= 0 by Def11;
  then
A2: |.((x + y).|.(x + y)).| >= 0 by Th29;
  sqrt ||.x + y.||^2 = sqrt |.((x + y).|.(x + y)).| by Th39,SQUARE_1:22;
  then
A3: ||.x + y.||^2 = |.((x + y).|.(x + y)).| by A2,A1,SQUARE_1:28;
  |.(y.|.y).| >= 0 by COMPLEX1:46;
  then
A4: |.(y.|.y).| = ||.y.||^2 by SQUARE_1:def 2;
A5: -Im(x.|.y) = Im((x.|.y)*') by COMPLEX1:27
    .= Im(y.|.x) by Def11;
  Im(x.|.x + x.|.y + y.|.x + y.|.y) = Im(x.|.x + x.|.y + y.|.x) + Im(y.|.y
  ) by COMPLEX1:8
    .= Im(x.|.x + x.|.y) + Im(y.|.x) + Im(y.|.y) by COMPLEX1:8
    .= Im(x.|.x) + Im(x.|.y) + Im(y.|.x) + Im(y.|.y) by COMPLEX1:8
    .= 0 + Im(x.|.y) + Im(y.|.x) + Im(y.|.y) by Def11
    .= Im(x.|.y) + Im(y.|.x) + 0 by Def11;
  then
A6: (x.|.x + x.|.y + y.|.x + y.|.y) = Re(x.|.x + x.|.y + y.|.x + y.|.y)+0*
  <i> by A5,COMPLEX1:13;
A7: Re(x.|.y) = Re((x.|.y)*') by COMPLEX1:27
    .= Re(y.|.x) by Def11;
A8: Re(x.|.y) <= |.(x.|.y).| by COMPLEX1:54;
  |.(x.|.y).| <= ||.x.||*||.y.|| by Th30;
  then Re(x.|.y) <= ||.x.||*||.y.|| by A8,XXREAL_0:2;
  then
A9: 2*Re(x.|.y) <= 2*(||.x.||*||.y.||) by XREAL_1:64;
  Re(x.|.x + x.|.y + y.|.x + y.|.y) = Re((x + y).|.(x + y)) by Th26;
  then
A10: Re(x.|.x + x.|.y + y.|.x + y.|.y) >= 0 by Def11;
  Re(x.|.x + x.|.y + y.|.x + y.|.y) = Re(x.|.x + x.|.y + y.|.x) + Re(y.|.y
  ) by COMPLEX1:8
    .= Re(x.|.x + x.|.y) + Re(y.|.x) + Re(y.|.y) by COMPLEX1:8
    .= Re(x.|.x) + Re(x.|.y) + Re(y.|.x) + Re(y.|.y) by COMPLEX1:8
    .= |.(x.|.x).| + Re(x.|.y) + Re(y.|.x) + Re(y.|.y) by Th29
    .= |.(x.|.x).| + Re(x.|.y) + Re(y.|.x) + |.(y.|.y).| by Th29;
  then |.(x.|.x + x.|.y + y.|.x + y.|.y).| = |.(x.|.x).| + 2*Re(x.|.y) + |.(y
  .|.y).| by A7,A10,A6,ABSVALUE:def 1;
  then
A11: ||.x + y.||^2 = 2*Re(x.|.y) + (|.(x.|.x).| + |.(y.|.y).|) by A3,Th26;
A12: ||.y.|| >= 0 by Th39;
  |.(x.|.x).| >= 0 by COMPLEX1:46;
  then (sqrt |.(x.|.x).|)^2 = |.(x.|.x).| by SQUARE_1:def 2;
  then
  ||.x + y.||^2 <= 2*(||.x.||*||.y.||) + (||.x.||^2 + |.(y.|.y).|) by A11,A9,
XREAL_1:6;
  then
A13: ||.x + y.||^2 <= (||.x.|| + ||.y.||)^2 by A4;
  ||.x.|| >= 0 by Th39;
  hence thesis by A12,A13,SQUARE_1:16;
end;
