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

theorem Lm21:
  for a,b,c be Real st a < b & b < c & b-a = c-b holds
    centroid (TriangularFS (a,b,c),['a,c']) = b
proof
 let a,b,c be Real;
 assume that
 A1: a < b & b < c and
 A2: b-a = c-b;
 A4: b-a > a-a by A1, XREAL_1:9;
 for x be Real holds TriangularFS (a,b,c).x = max(0,1-|.1*(x-b)/(b-a).|)
 proof
  let x be Real;
  max(0,1-|.(x-b)/(b-a).|)
  = TriangularFS (b-(b-a),b,b+(b-a)).x by FUZZY_5:65,A4;
  hence thesis by A2;
 end;
 then
 centroid (TriangularFS (a,b,c),['b - (b-a),b+(b-a)']) = b by Th19,A4;
 hence thesis by A2;
end;
