reserve x, y, z, v for set,
  n, m, k for Nat;
reserve G, G1, G2, G3 for Graph;
reserve x, y for Element of (the carrier of G);

theorem Th5:
  G1 c= G implies (the Source of G1) c= (the Source of G) &
  (the Target of G1) c= (the Target of G)
proof
  assume G1 c= G;
then A1: G1 is Subgraph of G;
 for v being object st v in (the Source of G1) holds v in (the Source of G)
  proof
    let v be object;
    assume
A2: v in (the Source of G1);
    then consider x, y being object such that
A3: [x,y] = v by RELAT_1:def 1;
A4: x in dom (the Source of G1) by A2,A3,FUNCT_1:1;
A5: y = (the Source of G1).x by A2,A3,FUNCT_1:1;
A6: x in the carrier' of G1 by A4;
 (the carrier' of G1) c= (the carrier' of G) by A1,Def18;
then  x in the carrier' of G by A6;
then A7: x in dom (the Source of G) by FUNCT_2:def 1;
 y = (the Source of G).x by A1,A4,A5,Def18;
    hence thesis by A3,A7,FUNCT_1:def 2;
  end;
  hence (the Source of G1) c= (the Source of G);
    let v be object;
    assume
A8: v in (the Target of G1);
    then consider x, y being object such that
A9: [x,y] = v by RELAT_1:def 1;
A10: x in dom (the Target of G1) by A8,A9,FUNCT_1:1;
A11: y = (the Target of G1).x by A8,A9,FUNCT_1:1;
A12: x in the carrier' of G1 by A10;
 (the carrier' of G1) c= (the carrier' of G) by A1,Def18;
then  x in the carrier' of G by A12;
then A13: x in dom (the Target of G) by FUNCT_2:def 1;
 y = (the Target of G).x by A1,A10,A11,Def18;
    hence thesis by A9,A13,FUNCT_1:def 2;
end;
