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 Th33:
  AffineMap(r,q`1,r,q`2).p = r*p+q
proof
  thus AffineMap(r,q`1,r,q`2).p = r*p+|[q`1,q`2]| by Th32
    .= r*p+q by EUCLID:53;
end;
