theorem ThI2:
  a ==> b implies a*c ==> b*c
  proof
    assume
A0: a ==> b;
    set o = In(3, dom the charact of S);
    arity Den(o, S) = 2 by ThB; then
    dom Den(o, S) = 2-tuples_on the carrier of S by MARGREL1:22; then
    reconsider ac = <*a,c*>, bc = <*b,c*> as Element of dom Den(o, S)
    by FINSEQ_2:101;
A2: dom <*a,c*> = Seg 2 & 1 in Seg 2 by FINSEQ_1:1,89;
A3: <*a,c*>.1 = a;
    <*a,c*>+*(1,b) = <*b,c*> by COMPUT_1:1; then
    Den(o,S).ac ==> Den(o,S).bc by A0,A2,A3,DEF2;
    hence a*c ==> b*c;
  end;
