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

theorem :: Theorem 3 (H)
  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
      UAp X = UAp X1 \/ UAp 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;
C1: UAp X c= UAp X1 \/ UAp X2
    proof
      let x be object;
      assume x in UAp X; then
      x in { x where x is Element of R :
      Class (the InternalRel of R, x) meets X } by ROUGHS_1:def 5; then
      consider xx being Element of R such that
A2:   xx = x & Class (the InternalRel of R, xx) meets X;
      consider z being object such that
A3:   z in Class (the InternalRel of R, xx) & z in X by A2,XBOOLE_0:3;
      reconsider zz = z as Element of R by A3;
      [xx,zz] in the InternalRel of R by A3,RELAT_1:169; then
B2:   [xx,zz] in the InternalRel of R1 or [xx,zz] in the InternalRel of R2
        by XBOOLE_0:def 3,A0;
      reconsider x1 = xx, z1 = zz as Element of R1 by A1,b1;
      reconsider x2 = xx, z2 = zz as Element of R2 by A1,b1;
      z1 in Class (the InternalRel of R1,x1) or
        z2 in Class (the InternalRel of R2,x2) by B2,RELAT_1:169; then
      Class (the InternalRel of R1,x1) meets X or
        Class (the InternalRel of R2,x2) meets X by A3,XBOOLE_0:3; then
      x1 in { x where x is Element of R1 :
        Class (the InternalRel of R1,x) meets X1 } or
      x2 in { x where x is Element of R2 :
        Class (the InternalRel of R2,x) meets X2 } by A1; then
      x1 in UAp X1 or x2 in UAp X2 by ROUGHS_1:def 5;
      hence thesis by A2,XBOOLE_0:def 3;
    end;
    UAp X1 \/ UAp X2 c= UAp X
    proof
      let x be object;
      assume x in UAp X1 \/ UAp X2; then
      per cases by XBOOLE_0:def 3;
      suppose
        x in UAp X1; then
        x in { x where x is Element of R1 :
          Class (the InternalRel of R1,x) meets X1 } by ROUGHS_1:def 5; then
        consider xx being Element of R1 such that
C1:     xx = x & Class (the InternalRel of R1,xx) meets X1;
        reconsider xxx = xx as Element of R by A1,b1;
        consider z being object such that
C2:     z in Class (the InternalRel of R1,xx) & z in X1 by C1,XBOOLE_0:3;
        reconsider zz = z as Element of R1 by C2;
        [xx,zz] in the InternalRel of R1 by C2,RELAT_1:169; then
        [xx,zz] in (the InternalRel of R1) \/ the InternalRel of R2
          by XBOOLE_0:def 3; then
        zz in Class (the InternalRel of R,xx) by A0,RELAT_1:169; then
        Class (the InternalRel of R,xx) meets X1 by C2,XBOOLE_0:3; then
        xxx in { x where x is Element of R :
          Class (the InternalRel of R,x) meets X } by A1;
        hence thesis by C1,ROUGHS_1:def 5;
      end;
      suppose
        x in UAp X2; then
        x in { x where x is Element of R2 :
          Class (the InternalRel of R2,x) meets X2 } by ROUGHS_1:def 5; then
        consider xx being Element of R2 such that
C1:     xx = x & Class (the InternalRel of R2,xx) meets X2;
        reconsider xxx = xx as Element of R by b1,A1;
        consider z being object such that
C2:     z in Class (the InternalRel of R2,xx) & z in X2 by C1,XBOOLE_0:3;
        reconsider zz = z as Element of R2 by C2;
        [xx,zz] in the InternalRel of R2 by C2,RELAT_1:169; then
        [xx,zz] in (the InternalRel of R1) \/ the InternalRel of R2
          by XBOOLE_0:def 3; then
        zz in Class (the InternalRel of R,xx) by A0,RELAT_1:169; then
        Class (the InternalRel of R,xx) meets X2 by C2,XBOOLE_0:3; then
        xxx in { x where x is Element of R :
          Class (the InternalRel of R,x) meets X } by A1;
        hence thesis by C1,ROUGHS_1:def 5;
      end;
    end;
    hence thesis by XBOOLE_0:def 10,C1;
  end;
