reserve a,b,s,t,u,lambda for Real,
  n for Nat;
reserve x,x1,x2,x3,y1,y2 for Element of REAL n;

theorem :: EUCLID_2:26
  for a,b being Real,x1,x2,x3 being Element of REAL n holds |( (a
  *x1+b*x2), x3 )| = a*|(x1,x3)| + b*|(x2,x3)|
proof
  let a,b be Real,x1,x2,x3 be Element of REAL n;
  thus |( (a*x1+b*x2), x3 )| = |(a*x1,x3)|+|(b*x2,x3)| by Th20
    .= a*|(x1,x3)|+|(b*x2,x3)| by Th21
    .= a*|(x1,x3)| + b*|(x2,x3)| by Th21;
end;
