theorem Th54:
  X c= P implies Sgm X = Sgm P * Sgm (Sgm P " X)
proof
  assume
A1: X c= P;
A2: Sgm P " X c= dom Sgm P by RELAT_1:132;
A3:  P is included_in_Seg;
A4: rng Sgm P=P by FINSEQ_1:def 14;
  rng ((Sgm P) | (Sgm P " X)) = (Sgm P).:(Sgm P " X) by RELAT_1:115
    .= X by A1,A4,FUNCT_1:77;
  hence thesis by A3,A2,FINSEQ_6:129;
end;
