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

theorem :: Theorem 4 (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 = Meet (R1,R2) & X = X1 & X = X2 &
    the carrier of R1 = the carrier of R2 holds
      UAp X c= 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 = Meet (R1,R2) & X = X1 & X = X2 &
    the carrier of R1 = the carrier of R2;
SS: the InternalRel of R =
      (the InternalRel of R1) /\ the InternalRel of R2 by A1,DefMeet;
sz: the carrier of R =
      (the carrier of R1) /\ the carrier of R2 by A1,DefMeet;
    reconsider XX1 = X as Subset of R1 by A1;
    reconsider XX2 = X as Subset of R2 by A1;
    reconsider XX = X as Subset of R;
zz: dom UAp R = bool the carrier of R by FUNCT_2:def 1;
    UAp X c= UAp X1 /\ UAp X2
    proof
      let x be object;
      assume
S2:   x in UAp X;
      UAp R cc= UAp R1 by Prop16H,SS,sz,A1,XBOOLE_1:17; then
      (UAp R).XX c= (UAp R1).XX1 by zz,ALTCAT_2:def 1; then
      (UAp R).XX c= UAp XX1 by ROUGHS_2:def 11; then
hh:   UAp XX c= UAp XX1 by ROUGHS_2:def 11;
      UAp R cc= UAp R2 by Prop16H,A1,SS,XBOOLE_1:17,sz; then
      (UAp R).XX c= (UAp R2).XX2 by zz,ALTCAT_2:def 1; then
      (UAp R).XX c= UAp XX2 by ROUGHS_2:def 11; then
      UAp XX c= UAp XX2 by ROUGHS_2:def 11;
      hence thesis by hh,XBOOLE_0:def 4,S2,A1;
    end;
    hence thesis;
  end;
