
theorem Th10:
  for P, R being Relation, x being object holds x in ["P,R"] iff [x`1
`1,x`2`1] in P & [x`1`2,x`2`2] in R &
 (ex a, b being object st x = [a,b]) &
 (ex c, d being object st x`1 = [c,d]) &
  ex e, f being object st x`2 = [e,f]
proof
  let P, R be Relation, x be object;
  hereby
    assume
A1: x in ["P,R"];
    then consider y, z being object such that
A2: x = [y,z] by RELAT_1:def 1;
    consider p,q,s,t being object such that
A3: y = [p,q] and
A4: z = [s,t] and
A5: [p,s] in P & [q,t] in R by A1,A2,Def1;
    x`1`1 = p & x`1`2 = q by A2,A3;
    hence [x`1`1,x`2`1] in P & [x`1`2,x`2`2] in R by A2,A4,A5;
    thus ex a, b being object st x = [a,b] by A2;
    thus ex c, d being object st x`1 = [c,d] by A2,A3;
    thus ex e, f being object st x`2 = [e,f] by A2,A4;
  end;
  assume that
A6: [x`1`1,x`2`1] in P and
A7: [x`1`2,x`2`2] in R;
  given a, b being object such that
A8: x = [a,b];
  given c, d being object such that
A9: x`1 = [c,d];
  given e, f being object such that
A10: x`2 = [e,f];
  [c,x`2`1] in P by A6,A9;
  then
A11: [c,e] in P by A10;
  [d,x`2`2] in R by A7,A9;
  then
A12: [d,f] in R by A10;
A13: b = [e,f] by A8,A10;
  a = [c,d] by A8,A9;
  hence thesis by A8,A13,A11,A12,Def1;
end;
