reserve A,X for non empty set;
reserve f for PartFunc of [:X,X:],REAL;
reserve a for Real;

theorem Th10:
  for R1,R2 being Relation of X holds R1 c= R2 implies R1[*] c= R2[*]
proof
  let R1,R2 be Relation of X;
  assume
A1: R1 c= R2;
A2: field R1 c= field R2 by A1,RELAT_1:16;
    let p be object such that
A3: p in R1[*];
    consider x,y being object such that
A4: p = [x,y] by A3,RELAT_1:def 1;
    consider r being FinSequence such that
A5: len r >= 1 & r.1 = x & r.(len r) = y and
A6: for i being Nat st i >= 1 & i < len r holds [r.i, r.(i+1)] in R1
    by A3,A4,FINSEQ_1:def 17;
A7: for i being Nat st i >= 1 & i < len r holds [r.i, r.(i+1)] in R2 by A1,A6;
    x in field R1 & y in field R1 by A3,A4,FINSEQ_1:def 17;
    hence p in R2[*] by A4,A5,A2,A7,FINSEQ_1:def 17;
end;
