
theorem
for a,b,p,q be Real, f be FuzzySet of REAL st
a > 0 & p > 0 & (-b)/a < q/p & (1-b)/a = (1-q)/(-p) &
 for x be Real holds
f.x
= max(0,min(1, ( (AffineMap (a,b) |].-infty,(q-b)/(a+p).[) +*
                 (AffineMap (-p,q)|[.(q-b)/(a+p),+infty.[) ) .x ))
holds f is triangular & f is strictly-normalized
proof
 let  a,b,p,q be Real, f be FuzzySet of REAL;
 assume that
 A1: a > 0 & p > 0 & (-b)/a < q/p & (1-b)/a = (1-q)/(-p) and
 A2: for x be Real holds
 f.x = max(0,min(1, ( (AffineMap (a,b) |].-infty,(q-b)/(a+p).[) +*
                 (AffineMap (-p,q)|[.(q-b)/(a+p),+infty.[) ) .x ));
 D1: REAL = dom f & REAL = dom (TriangularFS ((-b)/a,(1-b)/a,q/p))
       by FUNCT_2:52;
 for x being object st x in dom f holds
 (TriangularFS ((-b)/a,(1-b)/a,q/p)).x =f.x
 proof
  let x be object;
  assume x in dom f;
  then reconsider x as Real by D1;
  f.x = max(0,min(1, ( (AffineMap (a,b) |].-infty,(q-b)/(a+p).[) +*
                 (AffineMap (-p,q)|[.(q-b)/(a+p),+infty.[) ) .x )) by A2
  .= (TriangularFS ((-b)/a,(1-b)/a,q/p)).x by A1,asymTT6;
  hence thesis;
 end; then
 A4: f = (TriangularFS ((-b)/a,(1-b)/a,q/p)) by FUNCT_1:2,D1;
 0+(-q) < 1+(-q) & -p < 0 by A1, XREAL_1:8; then
 A7: (1-q)/(-p)< (-q)/(-p) by XREAL_1:75;
 0+(-b) < 1+(-b) by XREAL_1:8; then
  (-b)/a < (1-b)/a & (1-b)/a < q/p by XREAL_1:74,A7,A1,XCMPLX_1:191;
 hence thesis by FUZNUM_1:29,A4;
end;
