reserve x,y for object,X,Y,A,B,C,M for set;
reserve P,Q,R,R1,R2 for Relation;
reserve X,X1,X2 for Subset of A;
reserve Y for Subset of B;
reserve R,R1,R2 for Subset of [:A,B:];
reserve FR for Subset-Family of [:A,B:];

theorem :: (7.1.1)
  (union FR).:X = union {R.:X where R is Subset of [:A,B:]: R in FR}
proof
  thus
  (union FR).:X c= union {R.:X where R is Subset of [:A,B:]: R in FR}
  proof
    let a be object;
    assume a in (union FR).:X;
    then consider x being object such that
A1: [x,a] in union FR and
A2: x in X by RELAT_1:def 13;
    consider S being set such that
A3: [x,a] in S and
A4: S in FR by A1,TARSKI:def 4;
    reconsider S as Subset of [:A,B:] by A4;
    ex P being set st
    a in P & P in {T.:X where T is Subset of [:A,B:]: T in FR}
    proof
      set P = S.:X;
A5:   a in P by A2,A3,RELAT_1:def 13;
      P in {T.:X where T is Subset of [:A,B:]: T in FR} by A4;
      hence thesis by A5;
    end;
    hence thesis by TARSKI:def 4;
  end;
  let a be object;
  assume a in union {R.:X where R is Subset of [:A,B:]: R in FR};
  then consider P being set such that
A6: a in P and
A7: P in {R.:X where R is Subset of [:A,B:]: R in FR} by TARSKI:def 4;
  consider R being Subset of [:A,B:] such that
A8: P = R.:X and
A9: R in FR by A7;
  consider x being object such that
A10: [x,a] in R and
A11: x in X by A6,A8,RELAT_1:def 13;
  ex x being set st x in X & [x,a] in union FR
  proof
    take x;
    thus thesis by A9,A10,A11,TARSKI:def 4;
  end;
  hence thesis by RELAT_1:def 13;
end;
