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 Th32:
  AffineMap(r,s1,r,s2).p = r*p+|[s1,s2]|
proof
  thus AffineMap(r,s1,r,s2).p = |[r*(p`1)+s1,r*(p`2)+s2]| by JGRAPH_2:def 2
    .= |[(r*p)`1+s1,r*(p`2)+s2]| by TOPREAL3:4
    .= |[(r*p)`1+s1,(r*p)`2+s2]| by TOPREAL3:4
    .= |[(r*p)`1,(r*p)`2]|+ |[s1,s2]| by EUCLID:56
    .= r*p + |[s1,s2]| by EUCLID:53;
end;
