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 Th21:
  dist(q1+q,q2+q) = dist(q1,q2)
proof
A1: (q1+q)`1-(q2+q)`1 = q1`1+q`1-(q2+q)`1 by TOPREAL3:2
    .= q1`1+q`1-(q2`1+q`1) by TOPREAL3:2
    .= q1`1-q2`1+0;
A2: (q1+q)`2-(q2+q)`2 = q1`2+q`2-(q2+q)`2 by TOPREAL3:2
    .= q1`2+q`2-(q2`2+q`2) by TOPREAL3:2
    .= q1`2-q2`2+0;
  thus dist(q1+q,q2+q) = sqrt (((q1+q)`1-(q2+q)`1)^2 + ((q1+q)`2-(q2+q)`2)^2)
  by TOPREAL6:92
    .= dist(q1,q2) by A1,A2,TOPREAL6:92;
end;
