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 Th8:
  (p1+p2)`1+(p1+p2)`2*<i> = p1`1+p2`1+(p1`2+p2`2)*<i>
proof
A1: (p1+p2)=|[p1`1+p2`1 , p1`2+p2`2]| by EUCLID:55;
A2: Im (p1`1+p2`1+(p1`2+p2`2)*<i>)=p1`2+p2`2 by COMPLEX1:12;
A3: Im((p1+p2)`1+(p1+p2)`2*<i>)=(p1+p2)`2 & Re (p1`1+p2`1+(p1`2+p2`2)*<i>)=
  p1`1+ p2`1 by COMPLEX1:12;
  Re((p1+p2)`1+(p1+p2)`2*<i>)=(p1+p2)`1 by COMPLEX1:12;
  then Re((p1+p2)`1+(p1+p2)`2*<i>)=p1`1+p2`1 by A1,EUCLID:52;
  hence thesis by A1,A3,A2,EUCLID:52;
end;
