reserve a,b,c,d,x,y,z for object, X,Y,Z for set;
reserve R,S,T for Relation;
reserve F,G for Function;

theorem Th16:
  R is connected implies R |_2 Y is connected
proof
  assume
A1: R is connected;
  now
    let a,b;
    assume that
A2: a in field(R |_2 Y) & b in field(R |_2 Y) and
A3: a <> b;
    a in Y & b in Y by A2,Th12;
    then
A4: [a,b] in [:Y,Y:] & [b,a] in [:Y,Y:] by ZFMISC_1:87;
    a in field R & b in field R by A2,Th12;
    then [a,b] in R or [b,a] in R by A1,A3,Lm4;
    hence [a,b] in R |_2 Y or [b,a] in R |_2 Y by A4,XBOOLE_0:def 4;
  end;
  hence thesis by Lm4;
end;
