reserve X,X1,X2,Y,Y1,Y2 for set, p,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve f,g,g1,g2,h for Function,
  R,S for Relation;

theorem Th61:
  f is one-to-one implies f.:(X1 /\ X2) = f.:X1 /\ f.:X2
proof
  assume
A1: f is one-to-one;
A2: f.:X1 /\ f.:X2 c= f.:(X1 /\ X2)
  proof
    let y be object;
    assume
A3: y in f.:X1 /\ f.:X2;
    then y in f.:X1 by XBOOLE_0:def 4;
    then consider x1 being object such that
A4: x1 in dom f and
A5: x1 in X1 and
A6: y = f.x1 by Def6;
    y in f.:X2 by A3,XBOOLE_0:def 4;
    then consider x2 being object such that
A7: x2 in dom f and
A8: x2 in X2 and
A9: y = f.x2 by Def6;
    x1 = x2 by A1,A4,A6,A7,A9;
    then x1 in X1 /\ X2 by A5,A8,XBOOLE_0:def 4;
    hence thesis by A4,A6,Def6;
  end;
  f.:(X1 /\ X2)c=f.:X1 /\ f.:X2 by RELAT_1:121;
  hence thesis by A2;
end;
