theorem Th58:
  i1 <=> i2 in FI & j1 <=> j2 in FI implies (i1"\/"j1) <=> (i2"\/"
  j2) in FI & (i1"/\"j1) <=> (i2"/\"j2) in FI
proof
  assume that
A1: i1 <=> i2 in FI and
A2: j1 <=> j2 in FI;
A3: j1=>j2 in FI by A2,FILTER_0:8;
  then
A4: (i1"/\"j1)=>j2 in FI by Lm1;
A5: j1=>(i2"\/"j2) in FI by A3,Lm1;
A6: i1=>i2 in FI by A1,FILTER_0:8;
  then i1=>(i2"\/"j2) in FI by Lm1;
  then
A7: (i1"\/"j1) => (i2"\/"j2) in FI by A5,Lm2;
A8: j2=>j1 in FI by A2,FILTER_0:8;
  then
A9: (i2"/\"j2)=>j1 in FI by Lm1;
A10: i2=>i1 in FI by A1,FILTER_0:8;
  then (i2"/\"j2)=>i1 in FI by Lm1;
  then
A11: (i2"/\"j2) => (i1"/\"j1) in FI by A9,Lm3;
A12: j2=>(i1"\/"j1) in FI by A8,Lm1;
  i2=>(i1"\/"j1) in FI by A10,Lm1;
  then
A13: (i2"\/"j2) => (i1"\/"j1) in FI by A12,Lm2;
  (i1"/\"j1)=>i2 in FI by A6,Lm1;
  then (i1"/\"j1) => (i2"/\"j2) in FI by A4,Lm3;
  hence thesis by A11,A7,A13,FILTER_0:8;
end;
