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

theorem :: FUNCT_3:15
  for A,B being set, R being Subset of [:A,B:] holds
    union((.:R).:A) c= R.:(union A)
proof
  let A,B be set, R be Subset of [:A,B:];
  let y be object;
  assume y in union((.:R).:A);
  then consider Z being set such that
A1: y in Z and
A2: Z in (.:R).:A by TARSKI:def 4;
  consider X being object such that
A3: X in dom(.:R) and
A4: X in A and
A5: Z = (.:R).X by A2,FUNCT_1:def 6;
  reconsider X as set by TARSKI:1;
  y in R.:X by A1,A3,A5,Th19;
  then consider x being object such that
A6: [x,y] in R and
A7: x in X by RELAT_1:def 13;
  x in union A by A4,A7,TARSKI:def 4;
  hence thesis by A6,RELAT_1:def 13;
end;
