reserve x,y for object,X,Y,A,B,C,M for set;
reserve P,Q,R,R1,R2 for Relation;

theorem :: R(X) - original counterpart. The First Order Cutting.
  for A, B being set, X being Subset of A, R being Relation of A,B
  holds R.:X = union {Class(R,x) where x is Element of A: x in X}
proof
  let A, B be set, X be Subset of A, R be Relation of A,B;
  thus R.:X c= union {Class(R,x) where x is Element of A: x in X}
  proof
    let y be object;
    assume y in R.:X;
    then consider x1 being object such that
A1: [x1,y] in R and
A2: x1 in X by RELAT_1:def 13;
    x1 in union {{x} where x is Element of A: x in X} by A2,Th15;
    then consider S being set such that
A3: x1 in S and
A4: S in {{x} where x is Element of A: x in X} by TARSKI:def 4;
    consider x being Element of A such that
A5: S = {x} and
A6: x in X by A4;
A7: y in R.:{x} by A1,A3,A5,RELAT_1:def 13;
    set Y = R.:{x};
    Y = Class(R,x);
    then Y in {Class(R,xs) where xs is Element of A: xs in X} by A6;
    hence thesis by A7,TARSKI:def 4;
  end;
  let a be object;
  assume a in union {Class(R,x) where x is Element of A: x in X};
  then consider Y being set such that
A8: a in Y and
A9: Y in {Class(R,x) where x is Element of A: x in X} by TARSKI:def 4;
  consider x being Element of A such that
A10: Y = Class(R,x) and
A11: x in X by A9;
  Y c= R.:X
  proof
    let b be object;
    assume b in Y;
    then consider x1 being object such that
A12: [x1,b] in R and
A13: x1 in {x} by A10,RELAT_1:def 13;
    x1 = x by A13,TARSKI:def 1;
    hence thesis by A11,A12,RELAT_1:def 13;
  end;
  hence thesis by A8;
end;
