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 Th30:
  |( |[ x1, x2, x3 ]|, |[ y1, y2, y3 ]| )| = x1*y1 + x2*y2 + x3*y3
proof
  consider p1 such that
A1: p1 = |[x1, x2, x3]|;
A2: p1`3 = x3 by A1;
  consider p2 such that
A3: p2 = |[y1, y2, y3]|;
A4: p2`3 = y3 by A3;
A5: p2`1 = y1 & p2`2 = y2 by A3;
  p1`1 = x1 & p1`2 = x2 by A1;
  hence thesis by A1,A2,A3,A5,A4,Th29;
end;
