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 Th80:
  |.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;
A1: Rea (z1 * z2) = m1*n1 - m2*n2 - m3*n3 - m4*n4 by Lm17;
A2: Im1 (z1 * z2) = m1*n2 + m2*n1 + m3*n4 - m4*n3 by Lm17;
A3: Im2 (z1 * z2) = m1*n3 + m3*n1 + m4*n2 - m2*n4 by Lm17;
A4: Im3 (z1 * z2) = m1*n4 + m4*n1 + m2*n3 - m3*n2 by Lm17;
  set b1=m1*n1 - m2*n2 - m3*n3 - m4*n4, b2 = m1*n2 + m2*n1 + m3*n4 - m4*n3,
  b3 = m1*n3 + m3*n1 + m4*n2 - m2*n4, b4 = m1*n4 + m4*n1 + m2*n3 - m3*n2;
A5: (m1^2 + m2^2 + m3^2 + m4^2)>=0 by Th67;
A6: (n1^2 + n2^2 + n3^2 + n4^2) >= 0 by Th67;
  sqrt(b1^2 +b2^2 +b3^2 +b4^2)
  =sqrt((m1^2 + m2^2 + m3^2 + m4^2) * (n1^2 + n2^2 + n3^2 + n4^2));
  hence thesis by A1,A2,A3,A4,A5,A6,SQUARE_1:29;
end;
