reserve p1,p2,p3,p4,p5,p6,p,pc for Point of TOP-REAL 2;
reserve a,b,c,r,s for Real;
reserve c1,c2,c3 for Element of COMPLEX;

theorem
  c1 = euc2cpx(p1-p2) & c2 = euc2cpx(p3-p2) implies Re (c1.|.c2) = (p1`1
- p2`1)*(p3`1 - p2`1)+(p1`2 - p2`2)*(p3`2 - p2`2) & Im (c1.|.c2) = -(p1`1 - p2
`1)*(p3`2 - p2`2)+(p1`2 - p2`2)*(p3`1 - p2`1) & |.c1.| = sqrt((p1`1 - p2`1)^2 +
  (p1`2 - p2`2)^2) & |.p1-p2.|=|.c1.| by Lm16;
