
theorem Th1:
  for A,B being set, f being ManySortedSet of A for g being
  ManySortedSet of B st f c= g holds f# c= g#
proof
  let A,B be set, f be ManySortedSet of A;
  let g be ManySortedSet of B;
  assume
A1: f c= g;
A2: dom f c= dom g by A1,RELAT_1:11;
A3: dom g = B by PARTFUN1:def 2;
A4: dom (g# ) = B* by PARTFUN1:def 2;
    let z be object;
A5: dom f = A by PARTFUN1:def 2;
    assume
A6: z in f#;
    then consider x,y being object such that
A7: [x,y] = z by RELAT_1:def 1;
    dom (f# ) = A* by PARTFUN1:def 2;
    then reconsider x as Element of A* by A6,A7,XTUPLE_0:def 12;
A8: rng x c= A by FINSEQ_1:def 4;
    rng x c= A by FINSEQ_1:def 4;
    then rng x c= B by A5,A3,A2;
    then x is FinSequence of B by FINSEQ_1:def 4;
    then reconsider x9 = x as Element of B* by FINSEQ_1:def 11;
A9: rng x9 c= B by FINSEQ_1:def 4;
    y = f#.x by A6,A7,FUNCT_1:1
      .= product (f*x) by FINSEQ_2:def 5
      .= product (g*x9) by A1,A5,A3,A8,A9,PARTFUN1:54,79
      .= g#.x9 by FINSEQ_2:def 5;
    hence z in g# by A7,A4,FUNCT_1:1;
end;
