reserve a,b,c,x,y,z for object,X,Y,Z for set,
  n for Nat,
  i,j for Integer,
  r,r1,r2,r3,s for Real,
  c1,c2 for Complex,
  p for Point of TOP-REAL n;

theorem Th12:
  |.c1.| = |.c2.| & Arg c1 = Arg c2 + 2*PI*i implies c1 = c2
  proof
    assume
A1: |.c1.| = |.c2.| & Arg c1 = Arg c2 + 2*PI*i;
A2: cos Arg c2 = cos (Arg c2 + 2*PI*i) &
    sin Arg c2 = sin (Arg c2 + 2*PI*i) by COMPLEX2:8,9;
    c1 = |.c1.|*cos Arg c1 + |.c1.|*sin Arg c1 * <i> &
    c2 = |.c2.|*cos Arg c2 + |.c2.|*sin Arg c2 * <i> by COMPTRIG:62;
    hence c1 = c2 by A1,A2;
  end;
