 reserve R for 1-sorted;
 reserve X,Y for Subset of R;
 reserve R for finite 1-sorted;
 reserve X,Y for Subset of R;
 reserve R for finite Approximation_Space;
 reserve X,Y,Z,W for Subset of R;

theorem ::: Theorem 4.1 c) from Gomolinska's "A Comparative Study..."
  for R being finite Approximation_Space,
      u being Element of R,
      x,y being Subset of R st
   u in (f_1 R).x & (UncertaintyMap R).u = y
    holds kappa (y, x) > 0
  proof
    let R be finite Approximation_Space,
        u be Element of R,
        x,y be Subset of R;
    assume
AA: u in (f_1 R).x & (UncertaintyMap R).u = y;
    (f_1 R).x = { u where u is Element of R :
       (UncertaintyMap R).u meets x } by ROUGHS_5:def 5; then
    consider uu being Element of R such that
AB: uu = u & (UncertaintyMap R).uu meets x by AA;
    kappa (y,x) <> 0 by AB,AA,LemmaProp2b;
    hence thesis by XXREAL_1:1;
  end;
