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 Th41:
  for p,q holds |(p,q)| = p`1*q`1+p`2*q`2
proof
  let p,q;
  (p+q)`1=p`1+q`1 by TOPREAL3:2;
  then
A1: ((p+q)`1)^2=(p`1)^2+2*p`1*q`1+(q`1)^2 by SQUARE_1:4;
  (p+q)`2=p`2+q`2 by TOPREAL3:2;
  then
A2: ((p+q)`2)^2=(p`2)^2+2*p`2*q`2+(q`2)^2 by SQUARE_1:4;
  (p-q)`2=p`2-q`2 by TOPREAL3:3;
  then
A3: ((p-q)`2)^2=(p`2)^2-2*p`2*q`2+(q`2)^2 by SQUARE_1:5;
  (p-q)`1=p`1-q`1 by TOPREAL3:3;
  then
A4: ((p-q)`1)^2=(p`1)^2-2*p`1*q`1+(q`1)^2 by SQUARE_1:5;
  |(p,q)|= (1/4)*(|.p+q.|^2 - |.p-q.|^2) by EUCLID_2:49
    .= (1/4)*( ((p+q)`1)^2+((p+q)`2)^2 - |.p-q.|^2) by JGRAPH_3:1
    .= (1/4)*( ((p+q)`1)^2+((p+q)`2)^2 - (((p-q)`1)^2+((p-q)`2)^2)) by
JGRAPH_3:1
    .= (1/4)*( ((p+q)`1)^2 - ((p-q)`1)^2+(((p+q)`2)^2-((p-q)`2)^2));
  hence thesis by A1,A2,A4,A3;
end;
