reserve z,z1,z2 for Complex;
reserve r,x1,x2 for Real;
reserve p0,p,p1,p2,p3,q for Point of TOP-REAL 2;

theorem Th6:
  |[Re (z1+z2),Im (z1+z2)]|=|[Re z1 + Re z2, Im z1 + Im z2]|
proof
  (|[Re (z1+z2),Im (z1+z2)]|)`2=Im (z1 + z2) by EUCLID:52;
  then
A1: (|[Re (z1+z2),Im (z1+z2)]|)`2=Im z1 + Im z2 by COMPLEX1:8;
  (|[Re (z1+z2),Im (z1+z2)]|)`1=Re (z1+z2) by EUCLID:52;
  then (|[Re (z1+z2),Im (z1+z2)]|)`1=Re z1+Re z2 by COMPLEX1:8;
  hence thesis by A1,EUCLID:53;
end;
