 reserve X,a,b,c,x,y,z,t for set;
 reserve R for Relation;

theorem :: Theorem 3 (L)
  for R1, R2, R being non empty RelStr,
      X being Subset of R,
      X1 being Subset of R1,
      X2 being Subset of R2 st
    R = Union (R1,R2) & X = X1 & X = X2 &
    the carrier of R1 = the carrier of R2 holds
      LAp X = LAp X1 /\ LAp X2
  proof
    let R1, R2, R be non empty RelStr,
        X be Subset of R,
        X1 be Subset of R1,
        X2 be Subset of R2;
    assume
A1: R = Union (R1,R2) & X = X1 & X = X2 &
    the carrier of R1 = the carrier of R2; then
A0: the InternalRel of R = (the InternalRel of R1) \/ the InternalRel of R2
      by DefUnion;
b1: the carrier of R = (the carrier of R1) \/ the carrier of R2
      by A1,DefUnion;
D1: LAp X c= LAp X1 /\ LAp X2
    proof
      let x be object;
      assume x in LAp X; then
      x in { x where x is Element of R :
        Class (the InternalRel of R, x) c= X } by ROUGHS_1:def 4; then
      consider xx being Element of R such that
E1:   xx = x & Class (the InternalRel of R, xx) c= X;
      reconsider x1 = xx as Element of R1 by A1,b1;
      Class (the InternalRel of R1, x1) c= X1
      proof
        let y be object;
        assume
F1:     y in Class (the InternalRel of R1, x1); then
        reconsider y1 = y as Element of R1;
        [x1,y1] in the InternalRel of R1 by F1,RELAT_1:169; then
        [x1,y1] in (the InternalRel of R1) \/ (the InternalRel of R2)
          by XBOOLE_0:def 3; then
        y1 in Class (the InternalRel of R,xx) by A0,RELAT_1:169;
        hence thesis by E1,A1;
      end; then
      x1 in { x where x is Element of R1 :
        Class (the InternalRel of R1, x) c= X1 }; then
T1:   x1 in LAp X1 by ROUGHS_1:def 4;
      reconsider x2 = xx as Element of R2 by A1,b1;
      Class (the InternalRel of R2, x2) c= X2
      proof
        let y be object;
        assume
F1:     y in Class (the InternalRel of R2, x2); then
        reconsider y2 = y as Element of R2;
        [x2,y2] in the InternalRel of R2 by F1,RELAT_1:169; then
        [x2,y2] in (the InternalRel of R1) \/ (the InternalRel of R2)
          by XBOOLE_0:def 3; then
        y2 in Class (the InternalRel of R,xx) by A0,RELAT_1:169;
        hence thesis by E1,A1;
      end; then
      x2 in { x where x is Element of R2 :
        Class (the InternalRel of R2, x) c= X2 }; then
      x2 in LAp X2 by ROUGHS_1:def 4;
      hence thesis by T1,E1,XBOOLE_0:def 4;
    end;
    LAp X1 /\ LAp X2 c= LAp X
    proof
      let x be object;
      assume x in LAp X1 /\ LAp X2; then
H0:   x in LAp X1 & x in LAp X2 by XBOOLE_0:def 4; then
      x in { x where x is Element of R1 :
        Class (the InternalRel of R1, x) c= X1 } by ROUGHS_1:def 4; then
      consider x1 being Element of R1 such that
H1:   x1 = x & Class (the InternalRel of R1, x1) c= X1;
      x in { x where x is Element of R2 :
        Class (the InternalRel of R2, x) c= X2 } by H0,ROUGHS_1:def 4; then
      consider x2 being Element of R2 such that
H2:   x2 = x & Class (the InternalRel of R2, x2) c= X2;
      reconsider xx = x as Element of R by A1,b1,H1;
      Class (the InternalRel of R, xx) c= X
      proof
        let y be object;
        assume
S1:     y in Class (the InternalRel of R, xx); then
        reconsider yy = y as Element of R;
        [xx,yy] in the InternalRel of R by S1,RELAT_1:169; then
        [xx,yy] in (the InternalRel of R1) or
          [xx,yy] in the InternalRel of R2 by A0,XBOOLE_0:def 3; then
        yy in Class (the InternalRel of R1,xx) or
          yy in Class (the InternalRel of R2,xx) by RELAT_1:169;
        hence thesis by A1,H2,H1;
      end; then
      xx in { x where x is Element of R :
        Class (the InternalRel of R, x) c= X };
      hence thesis by ROUGHS_1:def 4;
    end;
    hence thesis by D1,XBOOLE_0:def 10;
  end;
