reserve X for set,
  Y for non empty set;
reserve n for Nat;
reserve r for Real,
  M for non empty MetrSpace;
reserve n for Nat,
  p,q,q1,q2 for Point of TOP-REAL 2,
  r,s1,s2,t1,t2 for Real,
  x,y for Point of Euclid 2;

theorem Th20:
  dist(|[0,0]|,r*q) = |.r.|*dist(|[0,0]|,q)
proof
A1: r^2 >= 0 & q`1^2 >=0 by XREAL_1:63;
A2: q`2^2 >=0 by XREAL_1:63;
A3: |[0,0]|`1 = 0 & |[0,0]|`2 = 0 by EUCLID:52;
  then
A4: dist(|[0,0]|,q) = sqrt((0-q`1)^2 + (0-q`2)^2) by TOPREAL6:92
    .= sqrt(q`1^2 + q`2^2);
  thus dist(|[0,0]|,r*q) = sqrt((0-(r*q)`1)^2 + (0-(r*q)`2)^2) by A3,
TOPREAL6:92
    .= sqrt(((r*q)`1)^2 + (-(r*q)`2)^2)
    .= sqrt((r*q`1)^2 + ((r*q)`2)^2) by TOPREAL3:4
    .= sqrt(r^2*q`1^2 + (r*q`2)^2) by TOPREAL3:4
    .= sqrt(r^2*(q`1^2 + q`2^2))
    .= sqrt(r^2)*sqrt(q`1^2 + q`2^2) by A1,A2,SQUARE_1:29
    .= |.r.|*dist(|[0,0]|,q) by A4,COMPLEX1:72;
end;
