 reserve A for non empty Subset of REAL;
 reserve A for non empty closed_interval Subset of REAL;

theorem
  for a,b,c,d be Real st a < b & b < c & b-a = c-b & d <> 0 holds
    centroid (d (#) TriangularFS (a,b,c),['a,c']) = b
proof
 let a,b,c,d be Real;
 assume that
 A1: a < b & b < c and
 A2: b-a = c-b and
 D1: d <> 0;
 TriangularFS (a,b,c) is_integrable_on ['a,c'] &
 TriangularFS (a,b,c) | ['a,c'] is bounded by Lm21L,A1; then
 centroid((d (#) TriangularFS (a,b,c)),['a,c'])
 = centroid(TriangularFS (a,b,c),['a,c']) by FUZZY_6:32,D1;
 hence thesis by Lm21,A1,A2;
end;
