reserve c,c1,c2 for Cardinal, G,G1,G2 for _Graph, v for Vertex of G;

theorem
  for X being non empty set, P being a_partition of X, c being Cardinal
  st for x being Element of X holds card EqClass(x,P) = c
  holds card X = c*`card P
proof
  let X be non empty set, P be a_partition of X, c be Cardinal;
  assume A1: for x being Element of X holds card EqClass(x,P) = c;
  for A, B being set st A in P & B in P & A <> B holds A misses B
    by EQREL_1:def 4;
  then A2: P is mutually-disjoint by TAXONOM2:def 5;
  now
    let A be set;
    set a = the Element of A;
    assume A3: A in P;
    then A4: A is Subset of X;
    A <> {} by A3, EQREL_1:def 4;
    then A5: a in A;
    then reconsider a as Element of X by A4;
    A = EqClass(a,P) by A3, A5, EQREL_1:def 6;
    hence card A = c by A1;
  end;
  then card union P = c*`card P by A2, Th56;
  hence thesis by EQREL_1:def 4;
end;
