reserve a,b,c,d,x,y,w,z,x1,x2,x3,x4 , X for set;
reserve A for non empty set;
reserve i,j,k for Element of NAT;
reserve a,b,c,d for Real;
reserve y,r,s,x,t,w for Element of RAT+;
reserve z,z1,z2,z3,z4 for Quaternion;
 reserve x for Real;

theorem Th72:
  |.z1 + z2.| <= |.z1.| + |.z2.|
proof
  set m1 = Rea z1, m2 = Im1 z1, m3 = Im2 z1, m4 = Im3 z1,
  n1 = Rea z2, n2 = Im1 z2, n3 = Im2 z2, n4 = Im3 z2, a = m1^2+m2^2+m3^2+m4^2,
  b = n1^2 +n2^2 + n3^2 + n4^2;
A1: |.z1.| >= 0 by Th60;
  |.z2.| >= 0 by Th60; then
A2: |.z1.| + |.z2.| >= 0 by A1;
A3: (sqrt (m1^2 + m2^2 + m3^2 + m4^2)+sqrt (n1^2 + n2^2 + n3^2 + n4^2))^2
  = (m1^2 + m2^2 + m3^2 + m4^2 + n1^2 + n2^2 + n3^2 + n4^2)+
  2*sqrt((m1^2 + m2^2 + m3^2 + m4^2) * (n1^2 + n2^2 + n3^2 + n4^2)) by Lm32;
A4: (sqrt ((m1 + n1)^2 + ( m2+n2)^2 + (m3+n3)^2 + (m4 +n4)^2))^2
  = (m1+n1)^2+(m2+n2)^2+(m3+n3)^2+(m4+n4)^2 by Lm33;
A5: Rea (z1+z2) = Rea z1 + Rea z2 by Th29;
A6: Im1 (z1+z2) = Im1 z1 + Im1 z2 by Th29;
A7: Im2 (z1+z2) = Im2 z1 + Im2 z2 by Th29;
A8: Im3 (z1+z2) = Im3 z1 + Im3 z2 by Th29;
A9: (m1*n2)^2 + (m2*n1)^2 >= 2*(m1*n2)*(m2*n1) by SERIES_3:6;
A10: (m1*n3)^2 + (m3*n1)^2 >= 2*(m1*n3)*(m3*n1) by SERIES_3:6;
A11: (m1*n4)^2 + (m4*n1)^2 >= 2*(m1*n4)*(m4*n1) by SERIES_3:6;
A12: (m2*n3)^2 + (m3*n2)^2 >= 2*(m2*n3)*(m3*n2) by SERIES_3:6;
A13: (m2*n4)^2 + (m4*n2)^2 >= 2*(m2*n4)*(m4*n2) by SERIES_3:6;
A14: (m3*n4)^2 + (m4*n3)^2 >= 2*(m3*n4)*(m4*n3) by SERIES_3:6;
  set a1= (m1*n2)^2, a2= (m2*n1)^2, a3=(m1*n3)^2,a4= (m3*n1)^2,
  a5= (m1*n4)^2, a6= (m4*n1)^2, a7= (m2*n3)^2, a8= (m3*n2)^2,
  a9 =(m2*n4)^2, a10= (m4*n2)^2, a11= (m3*n4)^2,a12= (m4*n3)^2,
  b1= 2*(m1*n2)*(m2*n1), b2= 2*(m1*n3)*(m3*n1),
  b3 = 2*(m1*n4)*(m4*n1), b4= 2*(m2*n3)*(m3*n2),
  b5= 2*(m2*n4)*(m4*n2), b6 =2*(m3*n4)*(m4*n3);
  ((a1 + a2) + (a3 + a4) + (a5 + a6)+(a7 + a8) + (a9 + a10) + (a11 + a12)) >=
  b1 + b2 + b3 + b4 + b5 + b6 by A9,A10,A11,A12,A13,A14,Lm30;
  then (a1 + a2 + a3 + a4 + a5 + a6+a7 + a8 + a9 + a10 + a11 + a12)
  +((m1*n1)^2+(m2*n2)^2+(m3*n3)^2+(m4*n4)^2) >= (b1 + b2 + b3 + b4 + b5 + b6)
  +((m1*n1)^2+(m2*n2)^2+(m3*n3)^2+(m4*n4)^2) by XREAL_1:6;
  then (a1 + a2 + a3 + a4 + a5 + a6+a7 + a8 + a9 + a10 + a11 + a12 )
  +((m1*n1)^2+(m2*n2)^2+(m3*n3)^2+(m4*n4)^2) >= (m1*n1+m2*n2+m3*n3+m4*n4)^2;
  then sqrt ((m1^2+m2^2+m3^2+m4^2)*(n1^2+n2^2+n3^2+n4^2)) >=
  m1*n1+m2*n2+m3*n3+m4*n4 by Lm29;
  then 2*sqrt ((m1^2+m2^2+m3^2+m4^2)*(n1^2+n2^2+n3^2+n4^2)) >=
  2*(m1*n1+m2*n2+m3*n3+m4*n4) by XREAL_1:64;
  then (a+b)+2*sqrt ((m1^2+m2^2+m3^2+m4^2)*(n1^2+n2^2+n3^2+n4^2)) >=
  (a+b)+2*(m1*n1+m2*n2+m3*n3+m4*n4) by XREAL_1:6;
  hence thesis by A2,A3,A4,A5,A6,A7,A8,Lm31;
end;
