
theorem Th15:
  for R being Relation, C, x, y being set holds x in SetBelow (R,C
  ,y) iff [x,y] in R & x in C
proof
  let R be Relation, C, x, y be set;
  hereby
    assume
A1: x in SetBelow (R,C,y);
    then x in R"{y} by XBOOLE_0:def 4;
    then ex a being object st [x,a] in R & a in {y} by RELAT_1:def 14;
    hence [x,y] in R by TARSKI:def 1;
    thus x in C by A1,XBOOLE_0:def 4;
  end;
  assume that
A2: [x,y] in R and
A3: x in C;
  y in {y} by TARSKI:def 1;
  then x in R"{y} by A2,RELAT_1:def 14;
  hence thesis by A3,XBOOLE_0:def 4;
end;
