reserve x,y,z for Real,
  x3,y3 for Real,
  p for Point of TOP-REAL 3;
reserve p1,p2,p3,p4 for Point of TOP-REAL 3,
  x1,x2,y1,y2,z1,z2 for Real;

theorem Th12:
  p1 - p2 = |[ p1`1 - p2`1, p1`2 - p2`2, p1`3 - p2`3]|
proof
A1: -p2 = |[ -p2`1, -p2`2, -p2`3]| by Th10;
  then
A2: (-p2)`3 = -p2`3;
  (-p2)`1 = -p2`1 & (-p2)`2 = -p2`2 by A1;
  hence p1 - p2 = |[ p1`1 + -p2`1, p1`2 + -p2`2, p1`3 + -p2`3]| by A2,Th5
    .= |[ p1`1 - p2`1, p1`2 - p2`2, p1`3 - p2`3]|;
end;
